Recent zbMATH articles in MSC 14https://zbmath.org/atom/cc/142022-09-13T20:28:31.338867ZUnknown authorWerkzeugBook review of: B. Poonen, Rational points on varietieshttps://zbmath.org/1491.000082022-09-13T20:28:31.338867Z"Baxa, C."https://zbmath.org/authors/?q=ai:baxa.christophReview of [Zbl 1387.14004].Book review of: L. Lovász, Graphs and geometryhttps://zbmath.org/1491.000162022-09-13T20:28:31.338867Z"Meunier, Frédéric"https://zbmath.org/authors/?q=ai:meunier.fredericReview of [Zbl 1425.05001].Giusto Bellavitis and its `Geometria di Derivazione'https://zbmath.org/1491.010202022-09-13T20:28:31.338867Z"Raspanti, Maria Anna"https://zbmath.org/authors/?q=ai:raspanti.maria-annaAuthor's abstract: Giusto Bellavitis (1803--1880), now mainly regarded for his theory of equipollences, was among the first Italian geometers, in 19th century, who devoted himself to the study of modern theories on projective transformations that were spreading among the transalpine geometers. His studies on geometric transformations -- to which he refers with the term of derivations, since they make it possible to derive the properties of one figure from those of another one -- are generally unknown. It is remarkable that Bellavitis, in \textit{Saggio di geometria derivata} (1838), besides having suggested the idea of quadric inversion, which would be studied about twenty years later by Thomas Archer Hirst (1830--1892), also provided a general definition of transfirmation, which is the same that Giovanni Virginio Schiaparelli (1835--1910) refers to in 1862 as conic transfirmation.
In this paper, Bellavitis' work is taken as a case-study of the importance, in the history of mathematics, to conceive concepts and theories within a context of scientific isolation, before their spreading within the scientific community. This paper also tries to shed light on the reasons why Bellavitis' works remained unknown for a long time. Finally, the paper tries to assess to what extent, some decades after their discovery, Bellavitis' works have contributed, although not decisively, to rethink, from an historical and critical standpoint, the development of the theory of geometric transformations.
Reviewer's remarks: One of the principal aims of the author is to unearth the value/importance of Bellavitis' research, set out in his paper of 1838. His work introduces the notion of ``equipollence. [Two line segments of equal length and also orientated in the same direction are called ``equipollent''. One recognizes here the notion of ``vector'', as later studied, for instance by Grassmann et al.] See also the book by \textit{M. J. Crowe} [A history of vector analysis. The evolution of the idea of a vectorial system. Notre Dame-London: University of Notre Dame Press (1967; Zbl 0165.00303)], in particular pp. 53--54. The author does not mention this important source in the references. He gives on pages 82 and 83 an example as worked out in Bellavitis' 1838 treatise. As a matter of fact, it concerns an earlier theorem due to Carnot (1803); it deals with three non-pairwise parallel lines intersected by means of an ellipse, and gives connections on the line segments involved inter alea. It is not clear whether Bellavitis knew of this earlier result; the author does not spend a word on it.
The list of the references is an eye opener. The interested reader should also consult the MacTutor History of Mathematics Archive as well as Wikipedia, concerning the merits of Bellavitis. The author did good work in order to brings to light the (relatively forgotten) credits of Bellavitis' research.
Reviewer: Robert W. van der Waall (Huizen)Maximum number of almost similar triangles in the planehttps://zbmath.org/1491.051372022-09-13T20:28:31.338867Z"Balogh, József"https://zbmath.org/authors/?q=ai:balogh.jozsef"Clemen, Felix Christian"https://zbmath.org/authors/?q=ai:clemen.felix-christian"Lidický, Bernard"https://zbmath.org/authors/?q=ai:lidicky.bernardSummary: A triangle \(T^\prime\) is \(\varepsilon\)-similar to another triangle \(T\) if their angles pairwise differ by at most \(\varepsilon\). Given a triangle \(T,\varepsilon > 0\) and \(n\in\mathbb{N}\), Bárány and Füredi asked to determine the maximum number of triangles \(h(n,T,\varepsilon)\) being \(\varepsilon\)-similar to \(T\) in a planar point set of size \(n\). We show that for almost all triangles \(T\) there exists \(\varepsilon=\varepsilon(T) > 0\) such that \(h(n,T,\varepsilon)=(1+o(1))n^3/24\). Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof.Packed words and quotient ringshttps://zbmath.org/1491.051892022-09-13T20:28:31.338867Z"Kroes, Daniël"https://zbmath.org/authors/?q=ai:kroes.daniel"Rhoades, Brendon"https://zbmath.org/authors/?q=ai:rhoades.brendonSummary: The coinvariant algebra is a quotient of the polynomial ring \(\mathbb{Q} [ x_1, \ldots, x_n]\) whose algebraic properties are governed by the combinatorics of permutations of length \(n\). A word \(w = w_1 \ldots w_n\) over the positive integers is packed if whenever \(i > 2\) appears as a letter of \(w\), so does \(i - 1\). We introduce a quotient \(S_n\) of \(\mathbb{Q} [ x_1, \ldots, x_n]\) which is governed by the combinatorics of packed words. We relate our quotient \(S_n\) to the generalized coinvariant rings of \textit{J. Haglund} et al. [Adv. Math. 329, 851--915 (2018; Zbl 1384.05043)] as well as the superspace coinvariant ring.Reduced word enumeration, complexity, and randomizationhttps://zbmath.org/1491.051902022-09-13T20:28:31.338867Z"Monical, Cara"https://zbmath.org/authors/?q=ai:monical.cara"Pankow, Benjamin"https://zbmath.org/authors/?q=ai:pankow.benjamin"Yong, Alexander"https://zbmath.org/authors/?q=ai:yong.alexanderSummary: A reduced word of a permutation \(w\) is a minimal length expression of \(w\) as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove that the Edelman-Greene statistic, defined by \textit{S. Billey} and \textit{B. Pawlowski} [J. Comb. Theory, Ser. A 127, 85--120 (2014; Zbl 1300.05313)], is typically exponentially large. This implies a result of \textit{B. Pawlowski} [Permutation diagrams in symmetric function theory and Schubert calculus. Seattle, WA: University of Washington (PhD Thesis) (2014), \url{https://digital.lib.washington.edu/researchworks/bitstream/handle/1773/26117/Pawlowski_washington_0250E_13742.pdf?sequence=1}], that it has exponentially growing expectation. Our result is established by a formal run-time analysis of Lascoux and Schützenberger's transition algorithm [\textit{A. Lascoux} and \textit{M.-P. Schützenberger}, Lett. Math. Phys. 10, 111--124 (1985; Zbl 0586.20007)]. The more general problem of Hecke word enumeration, and its closely related question of counting set-valued standard Young tableaux, is also investigated. The latter enumeration problem is further motivated by work on Brill-Noether varieties due to \textit{M. Chan} and \textit{N. Pflueger} [Trans. Am. Math. Soc. 374, No. 3, 1513--1533 (2021; Zbl 1464.14032)] and \textit{D. Anderson} et al. [``K-classes of Brill-Noether loci and a determinantal formula'', Int. Math. Res. Not. 2022, No. 16, 12653--12698 (2022; \url{doi:10.1093/imrn/rnab025})]. We also state some related problems about counting computational complexity.Hook formulas for skew shapes. IV: Increasing tableaux and factorial Grothendieck polynomialshttps://zbmath.org/1491.051932022-09-13T20:28:31.338867Z"Morales, A. H."https://zbmath.org/authors/?q=ai:morales.alejandro-h"Pak, I."https://zbmath.org/authors/?q=ai:pak.igor"Panova, G."https://zbmath.org/authors/?q=ai:panova.gretaSummary: We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes, our formulas generalize the classical hook-length formula and the Littlewood formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its \(q\)-analogs, which were studied in previous papers of the series.
For Part III see [the authors, Algebr. Comb. 2, No. 5, 815--861 (2019; Zbl 1425.05158)].Uncrowding algorithm for hook-valued tableauxhttps://zbmath.org/1491.051952022-09-13T20:28:31.338867Z"Pan, Jianping"https://zbmath.org/authors/?q=ai:pan.jianping"Pappe, Joseph"https://zbmath.org/authors/?q=ai:pappe.joseph"Poh, Wencin"https://zbmath.org/authors/?q=ai:poh.wencin"Schilling, Anne"https://zbmath.org/authors/?q=ai:schilling.anneA hook tableau is a semi-standard Young tableau shaped like an `L', in French notation. A hook-valued tableau is a tableau where each box contains a hook tableau, such that
\par i) if a box \(A\) is to the left of a box \(B\), but in the same row, then \(\max(A) \leqslant \min(B)\), and
\par ii) if a box \(A\) is below a box \(B\), but in the same column, then \(\max(A) < \min(C)\).
Here, \(\max(A)\) refers to the maximal entry of the hook tableau in box \(A\), and \(\min(A)\) is the minimal entry. Hook-valued tableaux generalise set-valued tableau and multiset-valued tableau: these result from the cases where the hooks consist of single columns and single rows, respectively. Just as with other sorts of tableaux, there is a crystal structure on hook-valued tableaux, introduced by \textit{G. Hawkes} and \textit{T. Scrimshaw} [Algebr. Comb. 3, No. 3, 727--755 (2020; Zbl 1441.05236)]. This specialises in the crystal structure of set-valued tableaux and multiset-valued tableaux.
For set-valued tableaux, there exists an uncrowding operator which maps a set-valued tableau to a pair consisting of a semi-standard Young tableau and a flagged increasing tableau. The operator is ``uncrowding'' in the sense that the output semistandard Young tableau has the same underlying multiset of numerical entries as the original set-valued tableau, except now we have one numerical entry per box, instead of a set of numerical entries per box. The flagged increasing tableau which is part of the output records data on how the tableau was uncrowded, thus allowing the original tableau to be reconstructed from the pair. A flagged increasing tableau is a tableau of skew shape which is increasing in both rows and columns such that entries in row \(i\) are at least \(i - 1\). An important property of the uncrowding operator on set-valued tableaux is that it intertwines with crystal operators.
The heart of the paper is the definition of an uncrowding operator for hook-valued tableaux. The output of such an operator is a set-valued tableaux and a column-flagged increasing tableaux. A column-flagged increasing tableau is the transpose of a flagged increasing tableau. This operator also has the desired property of intertwining with crystal operators. One can then uncrowded completely by uncrowding the output set-valued tableau.
There also exists an uncrowding operator on multiset-valued tableaux. The authors prove that their uncrowding operator on hook-valued tableaux generalises this operator. They also provide an inverse to their uncrowding map, giving a ``crowding'' map, which reassembles the original hook-valued tableau from a set-valued tableau and a column-flagged increasing tableau. This crowding map can only be applied to pairs that are compatible with each other in a certain way.
The authors also introduce an alternative uncrowding map on hook-valued tableaux which outputs a multiset-valued tableau and flagged increasing tableau. This uncrowding operator uncrowds the legs of the hooks in the hook-valued tableau, rather than the arms, as it were. It likewise intertwines with crystal operators.
In the final section, the authors apply their results to canonical Grothendieck polynomials. They use the uncrowding map to show that canonical Grothendieck polynomials have a tableau Schur expansion. Canonical Grothendieck polynomials are symmetric polynomials that can be expressed as generating functions of hook-valued tableaux. A symmetric function is said to have a tableau Schur expansion if it is the weighted sum of the Schur functions of a particular set of tableaux. A corollary of this result is an expansion of canonical Grothendieck polynomials in terms of stable symmetric Grothendieck polynomials and dual stable symmetric Grothendieck polynomials. Here, stable symmetric Grothendieck polynomials are generating functions of set-valued tableau and dual stable symmetric Grothendieck polynomials are generating functions of reverse plane partitions.
Reviewer: Nicholas Williams (Cologne)Double Grothendieck polynomials and colored lattice modelshttps://zbmath.org/1491.051962022-09-13T20:28:31.338867Z"Buciumas, Valentin"https://zbmath.org/authors/?q=ai:buciumas.valentin"Scrimshaw, Travis"https://zbmath.org/authors/?q=ai:scrimshaw.travisSummary: We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of \textit{A. Weigandt} [J. Comb. Theory, Ser. A 182, Article ID 105470, 52 p. (2021; Zbl 1475.05172)] relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of \textit{K. Motegi} and \textit{K. Sakai} [J. Phys. A, Math. Theor. 46, No. 35, Article ID 355201, 26 p. (2013; Zbl 1278.82042)] to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by \textit{P. J. McNamara} [Electron. J. Comb. 13, No. 1, Research paper R71, 40 p. (2006; Zbl 1099.05078)]. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.Commensurability in Mordell-Weil groups of abelian varieties and torihttps://zbmath.org/1491.110602022-09-13T20:28:31.338867Z"Banaszak, Grzegorz"https://zbmath.org/authors/?q=ai:banaszak.grzegorz"Blinkiewicz, Dorota"https://zbmath.org/authors/?q=ai:blinkiewicz.dorotaFrom the text: We investigate local to global properties for commensurability in Mordell-Weil groups of abelian varieties and tori via reduction maps.
In more detail, local to global properties for detecting linear relations in Mordell-Weil groups
of abelian varieties and tori have been investigated by numerous authors. Commensurability
questions in the Mordell-Weil groups have not yet been investigated in relation to reduction maps. In this paper we establish the relations between local to global detecting properties and local to global commensurability properties. We apply these results to Mordell-Weil groups of abelian varieties and tori. The structure of the paper is as follows. At the end of this introduction we define local to global commensurability properties. We also define notion of strong commensurability
in abelian groups with finite torsion. Then we define local to global properties for strong commensurability. In section 2 we investigate relations between local to global commensurability properties and local to global detecting properties. In section 3 we give examples of classes of abelian varieties and tori where the local to global strong commensurability property holds. In both cases we show examples of classes of abelian varieties and tori where the criterion fails.
As a corollary we obtain, in each case, four different Deligne 1-motives over a ring of integers, which become all equal to a torsion 1-motive, after base change and application of reduction map for almost all residue fields. In section 4 we give examples where one can check the strong commensurability in Mordell-Weil groups of abelian varieties and tori by finite number of reductions.Purely additive reduction of abelian varieties with torsionhttps://zbmath.org/1491.110612022-09-13T20:28:31.338867Z"Melistas, Mentzelos"https://zbmath.org/authors/?q=ai:melistas.mentzelosSummary: Let \(\mathcal{O}_K\) be a discrete valuation ring with fraction field \(K\) of characteristic 0 and algebraically closed residue field \(k\) of characteristic \(p > 0\). Let \(A / K\) be an abelian variety of dimension \(g\) with a \(K\)-rational point of order \(p\). In this article, we are interested in the reduction properties that \(A / K\) can have. After discussing the general case, we specialize to \(g = 1\), and we study the possible Kodaira types that can occur.Cycles in the de Rham cohomology of abelian varieties over number fieldshttps://zbmath.org/1491.110622022-09-13T20:28:31.338867Z"Tang, Yunqing"https://zbmath.org/authors/?q=ai:tang.yunqingSummary: In his 1982 paper, \textit{A. Ogus} [Lect. Notes Math. 900, 357--414 (1982; Zbl 0538.14010)] defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of \(\ell\)-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings' isogeny theorem due to Bost and the known cases of the Mumford-Tate conjecture.On a conjecture of Pappas and Rapoport about the standard local model for \(\mathrm{GL}_d\)https://zbmath.org/1491.110632022-09-13T20:28:31.338867Z"Muthiah, Dinakar"https://zbmath.org/authors/?q=ai:muthiah.dinakar"Weekes, Alex"https://zbmath.org/authors/?q=ai:weekes.alex"Yacobi, Oded"https://zbmath.org/authors/?q=ai:yacobi.odedLet \(n\) and \(e\) be integers greater than or equal to \(2\). Pappas and Rapoport conjectured that the subscheme \[\mathcal{N}_{n,e} = \{ A \in \mathrm{Mat}_{n \times n} \: : \: A^e = 0, \det(\lambda-A) = \lambda^n \}\] of the scheme \(\mathrm{Mat}_{n \times n}\) of \(n \times n\) matrices is reduced (Conjecture 5.8 of [\textit{G. Pappas} and \textit{M. Rapaport}, J. Algebraic Geom 12, 107--145 (2003; Zbl 1063.14029)]). They showed that this conjecture implies the flatness of the standard model of \(\mathrm{GL}_d\) in certain situations.
The above conjecture is proved in full generality in this paper.
Reviewer: Salman Abdulali (Greenville)On the existence of curves with prescribed \(a\)-numberhttps://zbmath.org/1491.110642022-09-13T20:28:31.338867Z"Zhou, Zijian"https://zbmath.org/authors/?q=ai:zhou.zijianLet \(k\) be an algebraically closed field of characteristic \(p>0\), let \(X\) be a curve defined over \(k\), and let \(\text{Jac}(X)\) be its Jacobian. One of the most important invariants of \(X\) is its \(a\)-number \(a_{X}\), which is defined by \[a_{X}=\text{dim}_{k}(\text{Hom}(\alpha_{p}, \text{Jac}(X))),\] where \(\alpha_{p}\) is the group scheme which is the kernel of Frobenius on the additive group scheme \(\mathbb{G}_{a}\). It is well known that the \(a\)-number of \(X\) is equal to \(g-r\), where \(g\) is the genus of \(X\) and \(r\) is the rank of the Cartier-Manin matrix, that is, the matrix for the Cartier operator defined on \(H^{0}(X,\Omega^{1}_{X})\).
In this paper, the author studies the existence of Artin-Schreier curves with large \(a\)-number, namely \(a_{X}=g-1\) and \(a_{X}=g-2\), proving among other important results that such curves can be written in some particular forms. Also, by computing the rank of the Hasse-Witt matrix of the curve, bounds on the \(a\)-number of trigonal curves of genus \(5\) in small characteristic are also given.
Reviewer: Mariana Coutinho (São Carlos)Explicit calculation of the mod 4 Galois representation associated with the Fermat quartichttps://zbmath.org/1491.110652022-09-13T20:28:31.338867Z"Ishitsuka, Yasuhiro"https://zbmath.org/authors/?q=ai:ishitsuka.yasuhiro"Ito, Tetsushi"https://zbmath.org/authors/?q=ai:ito.tetsushi"Ohshita, Tatsuya"https://zbmath.org/authors/?q=ai:ohshita.tatsuyaVisibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zerohttps://zbmath.org/1491.110662022-09-13T20:28:31.338867Z"Agashe, Amod"https://zbmath.org/authors/?q=ai:agashe.amodFrom the abstract: ``Let \(E\) be an optimal elliptic curve over \(\mathbb{Q}\) of conductor \(N\) having analytic rank zero, i.e., such that the \(L\)-function \(L_E(s)\) of \(E\) does not vanish at \(s = 1\). Suppose there is another optimal elliptic curve over \(\mathbb{Q}\) of the same conductor \(N\) whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to \(E\) modulo a power \(r\) of a prime \(p\). The theory of visibility then shows that under certain additional hypotheses involving \(p, r\) divides the product of the order of the Shafarevich-Tate group of \(E\) and the orders of the arithmetic component groups of \(E\). We extract an explicit integer factor from the Birch and Swinnerton-Dyer conjectural formula for the product mentioned above, and under some hypotheses similar to the ones made in the situation above, we show that \(r\) divides this integer factor. This provides theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank zero case.''
The author treats the case of the analytic rank zero while he previously obtained the case of the analytic rank one in [Int. Math. Res. Not. 2009, No. 15, 2899--2913 (2009; Zbl 1183.11035)] but the methods of two articles are different. One is the homology groups and the other is the Galois cohomology groups. Further, this article generalizes the case of \(r=p\) obtained in [the author, J. Reine Angew. Math. 644, 159--187 (2010; Zbl 1221.11147)].
Reviewer: Kazuma Morita (Sapporo)Cubic function fields with prescribed ramificationhttps://zbmath.org/1491.111002022-09-13T20:28:31.338867Z"Karemaker, Valentijn"https://zbmath.org/authors/?q=ai:karemaker.valentijn"Marques, Sophie"https://zbmath.org/authors/?q=ai:marques.sophie"Sijsling, Jeroen"https://zbmath.org/authors/?q=ai:sijsling.jeroenLet \(K\) be a function field with a perfect field \(k\) as its field of constants and let \(L/K\) be a separable geometric extension of degree \(3\). It is assumed that the characteristic of \(K\) is different from \(3\). Usually the construction or the study of these extensions are by means of the discriminant \(\delta\) of \(L/K\). In the paper under review, the authors classify extensions with a given set of ramified places instead of those with a given discriminant. The extension \(L/K\) is called {\em purely cubic} if there exists a generator \(y\) whose minimal polynomial over \(K\) is of the form \(X^3-\beta\), otherwise it is called {\em impurely cubic}. When \(L/K\) is impurely cubic, \(L/K\) admits a purely cubic closure \(K'\), which is a degree two extension of \(K\) such that \(LK'/K'\) is purely cubic.
The main result is Theorem 4.1, which establishes that if \(K=k(x)\) is a rational function field and if \(K'\) is of genus zero and \(T\) is a given set of places of \(K'\), then there exists a cubic extension \(L/K\) with purely cubic closure \(K'\) and triple ramification precisely at the places in \(T\) if and only if all places in \(T\) split in \(K'\).
The idea is to determine an extension \(L/K\) by descending from its base change \(LK'/K'\) to the purely cubic closure \(K'\).
In Section 6 the authors consider the case where \(K\) is not of genus \(0\) by constructing Parshin covers and applying the methods used for the case \(K=k(x)\). For cubic function fields of genus at most one, are described the twists and isomorphism classes obtained when Möbius transformations on \(K\) are allowed.
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)Moduli of local shtukas and Harris's conjecturehttps://zbmath.org/1491.111082022-09-13T20:28:31.338867Z"Hansen, David"https://zbmath.org/authors/?q=ai:hansen.davidThe paper under review deals with the cohomology of moduli space of mixed-characteristic local shtukas for \(\mathrm{GL}_n\). \textit{M. Rapoport} and \textit{Th. Zink} [Period spaces for \(p\)-divisible groups. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)] defined and studied the Rapopport-Zink spaces. These are local analogues of the Shimura varieties, and exist as rigid analytic varieties. As in the case of Shimura varieties, these spaces are expected to realize the local Langlands correspondance for reductive groups over local field. In his 2014 course at Berkeley, Scholze vastly generalized these spaces by constructing moduli spaces of mixed-characteristic local shtukas.
Let us fix a \(p\)-adic local field \(E\) with \([E: \mathbb{Q}_p]<\infty\). Let \(G\) be a split reductive group over \(E\) with fixed Borel \(\mathbf{B}\) over \(E\). Let \(\breve{E}\) be the completion of the maximal unramified extension of \(E\) and \(\mathbf{T}\subseteq \mathbf{B}\) be the maximal torus. We further fix a lift \(\sigma\in \text{Aut}(\breve{E}/E)\) of the \(q\)-Frobenius. The moduli space \(\text{Sht}_{G, \mu, b}\) depends on the group theoretic datum \((G, \mu, b)\), where \(\mu\in X_{\star}(\mathbf{T})_{\text{dom}}\) and \(b\in G(\breve{E})\) such that the \(\sigma\)-conjugacy class \([b]\in B(G)\) lies in the Kottwitz set \(B(G, \mu^{-1})\). This is a moduli space of local shtukas with one leg and infinite level structure. In general, the space \(\text{Sht}_{G, \mu, b}\) is not a rigid analytic variety and only a locally spatial diamond as defined in the Berkeley lectures on p-adic geometry. Note that a mixed-characteristic local shtukas can be interpreted as a \(G\)-bundle on the Fontaine-Fargues curve, which mimics the notion of a shtukas over function field by Drinfeld. The basic properties of \(\text{Sht}_{G, \mu, b}\) are summarized in Theorem 1.2 and proved in the case of \(G=\text{GL}_n\) in Section 2C.
Let \(C=\widehat{\bar{E}}\) and \(\text{ind}_{P}^G\) be the unnormalized parabolic induction. Following \textit{E. Mantovan} [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 5, 671--716 (2008; Zbl 1236.11101)], the author considers the so called Hodge-Newton parabolic \(\mathbf{P}=\mathbf{M}\mathbf{U}\) associated to the datum \((G, \mu, b)\) such that \(\mathbf{M}\subsetneq G\). In this case, the datum \((G, \mu, b)\) is called Hodge-Newton reducible (Definition 1.3). The main result (Theorem 1.8) of the paper states that we have canonical \(G\)-equivariant isomorphisms
\[
H_c^i(\text{Sht}_{G, \mu, b}\times_{\text{Spd} \breve{E}}\text{Spd} C, \mathbb{Z}/\ell^n)\simeq \text{ind}_{\mathbf{P}}^G(H_c^{i-2d}(\text{Sht}_{\mathbf{M}, \mu, b}\times_{\text{Spd} \breve{E}}\text{Spd} C, \mathbb{Z}/\ell^n)(-d))
\]
for all \(i\geq 0\) compatible with all additional structures, where
\[
d=\text{dim} \text{Sht}_{G, \mu, b}-\text{dim} \text{Sht}_{\mathbf{M}, \mu, b}.
\]
In particular, these isomorphisms are compatible with the natural \(W_E\)-actions on both sides. This settles a conjecture of Harris (in the Hodge-Newton reducible case) saying that when \(b\) is not basic, no supercuspidal representation of \(G\) contributes to the Euler characteristic of \(H_c^i(\text{Sht}_{G, \mu, b}, \overline{\mathbb{\mathbb{Q}}}_\ell)\).
Finally, the case of general group \(G\) is also settled in a preprint by \textit{I. Gaisin} and \textit{N. Imai} [``Non-semi-stable loci in Hecke stacks and Fargues' conjecture'', Preprint, \url{arXiv:1608.07446}].
Reviewer: Taiwang Deng (Bonn)Systems of polynomials with at least one positive real zerohttps://zbmath.org/1491.120012022-09-13T20:28:31.338867Z"Wang, Jie"https://zbmath.org/authors/?q=ai:wang.jie|wang.jie.2|wang.jie.1|wang.jie.3|wang.jie.4Splitting families in Galois cohomologyhttps://zbmath.org/1491.120022022-09-13T20:28:31.338867Z"Demarche, Cyril"https://zbmath.org/authors/?q=ai:demarche.cyril"Florence, Mathieu"https://zbmath.org/authors/?q=ai:florence.mathieuLet \(k\) be a field, \(k^s\) a separable closure of \(K\), \(\Gamma := \mathcal{G}(k^s/k)\) the absolute Galois group of \(k\), and \(p\) a prime number different from \(\operatorname{char}(k)\). Fix a finite étale group scheme \(A/k\) of multiplicative type, that is to say, a finite discrete \(\Gamma \)-module, and consider a class \(x \in H^n(k, A)\). The paper under review shows that there exists a countable family \((X_i)\), \(i \in I\), of smooth geometrically integral \(k\)-varieties, such that for any field extension \(\ell /k\) with \(\ell \) infinite, the restriction of \(x\) in \(H^n(\ell, A)\) vanishes if and only if \(X_i\) contains an \(\ell \)-point, for some \(i\). In addition, it proves that there exists such a family \((X_i)\) which is an ind-variety; by definition, this means that \(I = \mathbb{N}\) and, for each \(i \ge 0\), a closed embedding of \(k\)-varieties \(X_i \to X_{i+1}\) is given. For the proof, the authors consider separately the cases where \(n = 2\) and \(n \ge 3\). As a consequence of the main theorem in case \(n = 2\), they obtain that for each \(\alpha \in H^2(k, A)\), there is a smooth geometrically integral \(k\)-variety which is a splitting variety for \(\alpha \); this recover a result obtained by Krashen (Theorem~6.3 in: [\textit{D. Krashen}, Bull. Lond. Math. Soc. 48, No. 6, 985--1000 (2016; Zbl 1405.11039)].
Reviewer: Ivan D. Chipchakov (Sofia)Real Liouvillian extensions of partial differential fieldshttps://zbmath.org/1491.120042022-09-13T20:28:31.338867Z"Crespo, Teresa"https://zbmath.org/authors/?q=ai:crespo.teresa"Hajto, Zbigniew"https://zbmath.org/authors/?q=ai:hajto.zbigniew"Mohseni, Rouzbeh"https://zbmath.org/authors/?q=ai:mohseni.rouzbehAuthor's abstract: In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally \(p\)-adic differential fields with a \(p\)-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally \(p\)-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.
Reviewer: Michael F. Singer (Raleigh)Epsilon multiplicity for Noetherian graded algebrashttps://zbmath.org/1491.130022022-09-13T20:28:31.338867Z"Das, Suprajo"https://zbmath.org/authors/?q=ai:das.suprajoSummary: The notion of epsilon multiplicity was originally defined by \textit{B. Ulrich} and \textit{J. Validashti} [Math. Proc. Camb. Philos. Soc. 151, No. 1, 95--102 (2011; Zbl 1220.13006)] as a limsup, and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article, we show that the relative epsilon multiplicity of reduced Noetherian graded algebras over an excellent local ring exists as a limit. An important special case of \textit{S. D. Cutkosky}'s result [Adv. Math. 264, 55--113 (2014; Zbl 1350.13032), Theorem 6.3] concerning epsilon multiplicity is obtained as a corollary of our main theorem. We also produce a multigraded generalization of a result due to \textit{H. Dao} and \textit{J. Montaño} [Trans. Am. Math. Soc. 371, No. 5, 3483--3503 (2019; Zbl 1409.13036)] about monomial ideals.Virtual criterion for generalized Eagon-Northcott complexeshttps://zbmath.org/1491.130212022-09-13T20:28:31.338867Z"Booms-Peot, Caitlyn"https://zbmath.org/authors/?q=ai:booms-peot.caitlyn"Cobb, John"https://zbmath.org/authors/?q=ai:cobb.johnSummary: Given any map of finitely generated free modules, \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} defined a family of generalized Eagon-Northcott complexes associated to it [Adv. Math. 18, 245--301 (1975; Zbl 0336.13007)]. We give sufficient criterion for these complexes to be virtual resolutions, thus adding to the known examples of virtual resolutions, particularly those not coming from minimal free resolutions.The quantitative behavior of asymptotic syzygies for Hirzebruch surfaceshttps://zbmath.org/1491.130222022-09-13T20:28:31.338867Z"Bruce, Juliette"https://zbmath.org/authors/?q=ai:bruce.julietteIn the paper under review, the author studies the so-called Ein, Erman, and Lazarsfeld's normality heuristic [\textit{L. Ein} et al., J. Reine Angew. Math. 702, 55--75 (2015; Zbl 1338.13023)] for the asymptotic linear syzygies of Hirzebruch surfaces embedded by \(\mathcal{O}(d,2)\). More specifically, let \(X\) be projective variety of dimension \(n\) over an arbitrary field \(\mathbb{K}\). Given a sequence of very ample line bundles \(\{L_{d}\}_{d\in \mathbb{N}}\), one wants to study how the graded Betti numbers of \(X\) behave asymptotically with respect to \(L_{d}\) if \(d \gg 0\), that is, one is interested in the syzygies of the section ring \[R(X;L_{d}) :=\bigoplus_{k \in \mathbb{Z}}H^{0}(X,kL_{d})\] as a module over \(S = \mathrm{Sym} \, H^{0}(X,L_{d})\). Considering the graded minimal free resolution \[0 \rightarrow F_{r_{d}} \rightarrow \cdots \quad \cdots \rightarrow F_{1} \rightarrow F_{0} \rightarrow R(X;L_{d}) \rightarrow 0,\] let \[K_{p,q}(X;L_{d}) := \mathrm{span}_{\mathbb{K}} \langle\text{minimal generators of } \, F_{p} \, \text{ of degree } \, (p+q)\rangle\] be the finite dimensional \(\mathbb{K}\)-vector space of minimal syzygies of homological degree \(p\) and degree \(p+q\). We write \(k_{p,q}(X;L_{d})\) for \(\mathrm{dim} \, K_{p,q}(X;L_{d})\), and then form the Betti table of \((X;L_{d})\) by placing \(k_{p,q}(X;L_{d})\) in the \((q,p)\)-th spot.
In this setup, we can state Ein, Erman, and Lazarsfeld's heuristic as follows: if \(\{L_{d}\}_{d\in \mathbb{N}}\) is a sequence of line bundles growing in positivity, then for any \(q \in [1, ..., n]\) there exists a function \(F_{q}(d)\), depending on \(X\), such that if \(\{p_{d}\}_{d\in \mathbb{N}}\) is a sequence of non-negative integers such that \[(\star): \quad \quad \mathrm{lim}_{d \rightarrow \infty}\bigg(p_{d} - (r_{d}/2 + a\sqrt{r_{d}}/2)\bigg) = 0,\] where \(a \in \mathbb{R}\) is a fixed constant, then \[F_{q}(d) \cdot k_{p_{d},q}(X;L_{d}) \rightarrow e^{-a^{2}/2}.\]
Now we can formulate the main results of the paper.
Denote by \(\mathbb{F}_{t}\) the Hirzebruch surface embedded by the line bundle \(\mathcal{O}_{\mathbb{F}_{t}}(d,2)\).
Theorem A. If \(\{p_{d}\}_{d \in \mathbb{N}}\) is a sequence of non-negative integers satisfying \((\star)\) for some real number \(a \in \mathbb{R}\), then \[\frac{3\sqrt{2\pi}}{2^{r_{d}}\sqrt{r_{d}}} \cdot k_{p_{d},1}(\mathbb{F}_{t}, \mathcal{O}_{\mathbb{F}_{t}}(d,2)) = e^{-a^{2}/2}\bigg(1 + O\bigg(\frac{1}{\sqrt{r_{d}}}\bigg)\bigg).\]
Theorem B. There does not exist a function \(F_{2}(d)\) such that if \(\{p_{d}\}_{d\in \mathbb{N}}\) is a sequence of non-negative integers satisfying \((\star)\) for some real number \(a \in \mathbb{R}\), then \[F_{2}(d)\cdot k_{p_{d},2}(\mathbb{F}_{t},\mathcal{O}_{\mathbb{F}_{t}}(d,2)) = e^{-a^{2}/2}\bigg(1 + O\bigg(\frac{1}{\sqrt{r_{d}}}\bigg)\bigg).\]
Reviewer: Piotr Pokora (Kraków)On the stable under specialization sets and cofiniteness of local cohomologyhttps://zbmath.org/1491.130252022-09-13T20:28:31.338867Z"Aghapournahr, Moharram"https://zbmath.org/authors/?q=ai:aghapournahr.moharram"Hatamkhani, Marziye"https://zbmath.org/authors/?q=ai:hatamkhani.marziyeThe paper under review deals with a generalization of the concept of cofiniteness defined in [\textit{R. Hartshorne}, Invent. Math. 9, 145--164 (1970; Zbl 0196.24301)] with a view using the so-called stable under specialization defined in [\textit{K. Divaani-Aazar} et al., J. Algebra Appl. 18, No. 1, Article ID 1950015, 22 p. (2019; Zbl 1454.13025)].
Reviewer: Majid Eghbali (Tehran)Frobenius-Witt differentials and regularityhttps://zbmath.org/1491.130312022-09-13T20:28:31.338867Z"Saito, Takeshi"https://zbmath.org/authors/?q=ai:saito.takeshiIn [J. Algebra 524, 110--123 (2019; Zbl 1408.13054)], the authors introduced the module of total \(p\)-differentials for a ring over \(\mathbb{Z}/p^{2}\). The paper under review studies a similar construction for a ring over \(\mathbb{Z}_{(p)}\). Given a (commutative) ring \(R\), an \(R\)-module \(M\) and a prime number \(p\), the paper defines a Frobenius-Witt derivation (or a FW-derivation) from \(A\) to \(M\) as a mapping \(w:A\rightarrow M\) such that \(w(a+b) = w(a) + w(b) - P(a, b)w(p)\) and \(w(ab) = b^{p}w(a)+a^{p}w(b)\) where \(P = \sum_{i=1}^{p-1}{\frac{(p-1)!}{i!(p-i)!}}X^{i}Y^{p-i}\in\mathbb{Z}[X, Y]\). It is shown that there exists a universal pair of an \(A\)-module \(F\Omega_{A}^{1}\) and an FW-derivation \(w:A\rightarrow F\Omega_{A}^{1}\) called the \textit{module of} FW-\textit{differentials} of \(A\) and the \textit{universal} FW-\textit{derivation}, respectively. The main result of the paper under review is a proof of a regularity criterion in this case. It says that under a suitable finitness condition, a Noetherian local ring \(A\) with residue field \(k\) of characteristic \(p\) is regular if and only if the \(A/pA\)-module of FW-differentials \(F\Omega_{A}^{1}\) is free of rank \(d+r\) where \(d = \dim A\) and \([k:k^{p}] = p^{r}\). The construction of \(F\Omega_{A}^{1}\) is sheafified and the author obtains a sheaf of FW-differentials \(F\Omega_{X}^{1}\) on a scheme \(X\). It is used in [\textit{T. Saito}, Algebra Number Theory 16, No. 2, 335--368 (2022; Zbl 07516272)] to define the cotangent bundle and the microsupport of an étale sheaf in mixed characteristic. The last part of the paper under review studies the relation of \(F\Omega_{X}^{1}\) with \(\mathcal{H}_{1}\) of cotangent complexes.
Reviewer: Alexander B. Levin (Washington)On exponential morphisms over commutative ringshttps://zbmath.org/1491.130322022-09-13T20:28:31.338867Z"El Kahoui, M'hammed"https://zbmath.org/authors/?q=ai:el-kahoui.mhammed"Hammi, Aziza"https://zbmath.org/authors/?q=ai:hammi.azizaSummary: We give elementary and self-contained proofs of two results concerning exponential morphisms on polynomial rings.Two algorithms for computing the general component of jet scheme and applicationshttps://zbmath.org/1491.130362022-09-13T20:28:31.338867Z"Cañón, Mario Morán"https://zbmath.org/authors/?q=ai:canon.mario-moran"Sebag, Julien"https://zbmath.org/authors/?q=ai:sebag.julienLet \(X\) be an integral variety over a perfect field \(k\), \(\mathcal L_m(X)\) its jet scheme of level \(m\in \mathbb N\) and \(\mathcal L_{\infty}(X)\) its arc scheme. The general component \(\mathcal G_m(X)\) of \(\mathcal L_m(X)\) is the Zariski closure of \(\mathcal L_m(\mathrm{Reg}(X))\).
If \(X\) is smooth on \(k\) then the geometry and topology of \(\mathcal L_m(X)\) are well understood. In this paper the authors consider the case \(X\) is not smooth and study some properties of the general component \(\mathcal G_m(X)\) by means of a smooth birational model of \(X\).
Indeed, under the further hypothesis that \(X\) is affine embedded in \(\mathbb A^N_k\), the authors prove that a birational model of \(X\) provides a description of \(\mathcal G_m(X)\) that gives rice to an algorithm which computes a Groebner basis of the defining ideal of \(\mathcal G_m(X)\) in \(\mathbb A^N_k\) as a subscheme of \(\mathcal L_m(X)\) (Algorithm~2). The authors also extend to arbitrary integral varieties over perfect fields over arbitrary characteristic another algorithm ''already introduced in the Ph.D. Thesis of Kpognon'' (see also [\textit{K. Kpognon} and \textit{J. Sebag}, Commun. Algebra 45, No. 5, 2195--2221 (2017; Zbl 1376.14018)]) ``for the study of arc scheme associated with integral affine plane curves in characteristic zero'' (Algorithm~1). Several examples and comments to the implementation of the algorithms, which is available in SageMath, are provided in Sections~6 and~7.
The given results are applied for further studies of plane curves, concerning differential operators logarithmic along an affine plane curve and the rationality of a motivic power series that is introduced by the authors and ``which encodes the geometry of all \(\mathcal G_m(X)\)'' (Sections 8 and 9).
Reviewer: Francesca Cioffi (Napoli)Representation theory and algebraic geometry. A conference celebrating the birthdays of Sasha Beilinson and Victor Ginzburg, Chicago, IL, USA, August 21--25, 2017https://zbmath.org/1491.140012022-09-13T20:28:31.338867ZPublisher's description: The chapters in this volume explore the influence of the Russian school on the development of algebraic geometry and representation theory, particularly the pioneering work of two of its illustrious members, Alexander Beilinson and Victor Ginzburg, in celebration of their 60th birthdays. Based on the work of speakers and invited participants at the conference ``Interactions Between Representation Theory and Algebraic Geometry'', held at the University of Chicago, August 21--25, 2017, this volume illustrates the impact of their research and how it has shaped the development of various branches of mathematics through the use of D-modules, the affine Grassmannian, symplectic algebraic geometry, and other topics. All authors have been deeply influenced by their ideas and present here cutting-edge developments on modern topics. Chapters are organized around three distinct themes:
\begin{itemize}
\item Groups, algebras, categories, and representation theory
\item \(D\)-modules and perverse sheaves
\item Analogous varieties defined by quivers
\end{itemize}
Representation Theory and Algebraic Geometry will be an ideal resource for researchers who work in the area, particularly those interested in exploring the impact of the Russian school.
The articles of this volume will be reviewed individually.Seshadri's criterion and openness of projectivityhttps://zbmath.org/1491.140022022-09-13T20:28:31.338867Z"Kollár, János"https://zbmath.org/authors/?q=ai:kollar.janosSummary: We prove that projectivity is an open condition for deformations of algebraic spaces with rational singularities.Equisingularity of families of functions on isolated determinantal singularitieshttps://zbmath.org/1491.140032022-09-13T20:28:31.338867Z"Carvalho, R. S."https://zbmath.org/authors/?q=ai:carvalho.r-s"Nuño-Ballesteros, J. J."https://zbmath.org/authors/?q=ai:nuno-ballesteros.juan-jose"Oréfice-Okamoto, B."https://zbmath.org/authors/?q=ai:orefice-okamoto.b"Tomazella, J. N."https://zbmath.org/authors/?q=ai:tomazella.joao-nivaldoIn [\textit{J. J. Nuño-Ballesteros} et al., Math. Z. 289, No. 3--4, 1409--1425 (2018; Zbl 1400.32015)], some of the authors showed that the family of varieties \(\left \{(X_t, 0) \right \}_{t \in D}\) is Whitney equisingular if and only if it is good and all the polar multiplicities \(m_i(X_t, 0)\), \(i = 0, \dots, d\) are constant on \(t\).
In this paper, the authors also characterize the Whitney equisingularity for analytic families of function germs \(F=\left \{f_t : (X_t, 0) \to (\mathbb{C}, 0)\right \}_{t \in D}\) with isolated critical points, where \((X_t, 0)\) are \(d\)-dimensional isolated determinantal singularities.
For this, the authors introduce the \((d-1)\)th polar multiplicity of the fiber \(Y:=f^{-1}(0) \subset X\) of a function germ \(f:(X,0) \to (\mathbb{C},0)\) with isolated singularity. The main result is: the family \(F\) is Whitney equisingular if only if \((\mathcal{X},0)=\left \{(X_t, 0) \right \}_{t \in D}\) is a good family, \(m_i(X_t,0)\), \(i=0, \dots, d\) and \(m_k(Y_t,0)\), \(k=0,\dots, d-1\) are constant on \(t\in D\).
Reviewer: Daiane Alice Henrique Ament (Lavras)Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularitieshttps://zbmath.org/1491.140042022-09-13T20:28:31.338867Z"Hussain, Naveed"https://zbmath.org/authors/?q=ai:hussain.naveed"Yau, Stephen S.-T."https://zbmath.org/authors/?q=ai:yau.stephen-shing-toung"Zuo, Huaiqing"https://zbmath.org/authors/?q=ai:zuo.huaiqingIt is well known that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but solvable and nilpotent Lie algebras are not. In the paper under review, the authors give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. In particular, they get the correspondence between the nilpotent Lie algebras of dimension less than or equal to \(7\) and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to \(1\).
Reviewer: Rong Du (Shanghai)Multiplicity, regularity and blow-spherical equivalence of real analytic setshttps://zbmath.org/1491.140052022-09-13T20:28:31.338867Z"Sampaio, José Edson"https://zbmath.org/authors/?q=ai:sampaio.j-edsonThe author is interested in studying multiplicity and regularity on an analytic setting and, in this context, he introduces a weaker variation of the concept of blow-spherical equivalence introduced by \textit{A. Fernandes} et al. [Indiana Univ. Math. J. 66, No. 2, 547--557 (2017; Zbl 1366.14051)], which he also names blow-spherical equivalence or equivalence under a blow-spherical homeomorphism. Equivalence under this concept lives strictly between topological equivalence and sub analytic bi-Lipschitz equivalence and, also, between topological equivalence and differential equivalence.
Recently, the author conjectured that if \(X\) and \(Y\) are two real analytic sets included in \(\mathbb{R}^n \) and there is a bi-Lipschitz homeomorphism: \( \varphi: (\mathbb{R}^n, X, 0) \rightarrow (\mathbb{R}^n, Y, 0)\), then the multiplicities at origin of \(X\) and \(Y\) satisfy \(m(X) \equiv m(Y) \bmod 2\). He proved that this conjecture is true when \(n=3\). In this paper, the author proposes the same conjecture but replacing ``bi-Lipschitz'' with ``blow-spherical''. Proposition 5.2, Theorems 5.5, 5.7 and 5.16 and Corollary 5.18 in the paper prove that the conjecture holds whenever \(n \leq 3\) or whenever \(\varphi\) is also image arc-analytic. The author also gives a real analogue of the Gau-Lipman's Theorem.
Finally, and concerning regularity of complex analytic sets, the author proves that if a real analytic set \(X \subseteq \mathbb{R}^n\) is blow-spherical regular at \(0 \in X\), then \(X\) is \(C^1\) smooth at \(0\) if and only if the dimension \(d\) of \(X\) is \(1\).
Reviewer: Carlos Galindo (Castellón)Dimension of the moduli space of a germ of curve in \(\mathbb{C}^2\)https://zbmath.org/1491.140062022-09-13T20:28:31.338867Z"Genzmer, Yohann"https://zbmath.org/authors/?q=ai:genzmer.yohannThe paper studies the moduli space \(\mathbb{M}(S) := \mathrm{Top}(S,0)/\mathrm{Diff}(\mathbb{C}^{2},0)\) of the topological class of a curve germ \((S, 0)\subset (\mathbb{C}^{2},0)\) by the action of the group \(\mathrm{Diff}(\mathbb{C}^{2},0)\). The authors prove an explicit formula for the generic dimension of the moduli space in terms of the minimal resolution of the curve S, by using technics from the theory of holomorphic foliations.
Reviewer: Mihai-Marius Tibar (Lille)On the algebraicity about the Hodge numbers of the Hilbert schemes of algebraic surfaceshttps://zbmath.org/1491.140072022-09-13T20:28:31.338867Z"Jin, Seokho"https://zbmath.org/authors/?q=ai:jin.seokho"Jo, Sihun"https://zbmath.org/authors/?q=ai:jo.sihunSummary: Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of \(n\) points of an algebraic surface is algebraic at a CM point \(\tau\) and rational numbers \(z_1\) and \(z_2\). Our result gives a refinement of the algebraicity on Betti numbers.A-Hilbert schemes for \(\displaystyle\frac{1}{r}(1^{n-1},a)\)https://zbmath.org/1491.140082022-09-13T20:28:31.338867Z"Jung, Seung-Jo"https://zbmath.org/authors/?q=ai:jung.seung-joSummary: For a finite group \(G \subset \mathrm{GL}(n, \mathbb{C})\), the \(G\)-Hilbert scheme is a fine moduli space of \(G\)-clusters, which are 0-dimensional \(G\)-invariant subschemes \(Z\) with \(H^0(\mathcal{O}_Z)\) isomorphic to \(\mathbb{C}[G]\). In many cases, the \(G\)-Hilbert scheme provides a good resolution of the quotient singularity \(\mathbb{C}^n/G\), but in general it can be very singular. In this note, we prove that for a cyclic group \(A \subset \mathrm{GL}(n, \mathbb{C})\) of type \(\frac{1}{r}(1, \dots, 1, a)\) with \(r\) coprime to \(a\), \(A\)-Hilbert Scheme is smooth and irreducible.Lehn's formula in Chow and conjectures of Beauville and Voisinhttps://zbmath.org/1491.140092022-09-13T20:28:31.338867Z"Maulik, Davesh"https://zbmath.org/authors/?q=ai:maulik.davesh"Neguţ, Andrei"https://zbmath.org/authors/?q=ai:negut.andreiIn this paper, motivated by the Beauville-Voisin conjecture, the authors study the cycle class map of the Hilbert schemes of points on a \(K3\) surface.
The Beauville-Voisin conjecture for a hyperkähler manifold \(X\) states that the subring of the Chow ring generated by the divisor classes and the Chern characters of the tangent bundle injects into the cohomology ring of \(X\). Let \(S\) be a \(K3\) surface over an algebraically closed field of characteristic \(0\). It is known that the Hilbert scheme \(\mathrm{Hilb}_n(S)\) of \(n\) points on \(S\) is hyperkähler.
Let \(A^*(\cdot)\) and \(H^*(\cdot)\) denote the Chow ring and the cohomology ring with \(\mathbb Q\)-coefficients respectively. A small tautological class is defined to be any element of \(A^*(\mathrm{Hilb}_n(S))\) of the form \( \pi_{1*}(\mathrm{ch}_k(\mathcal{O}_{\mathcal{Z}_n}) \cdot \pi_2^*\gamma) \) where \(\mathcal{Z}_n\) is the universal subscheme of \(\mathrm{Hilb}_n(S) \times S\), \(\pi_1\) and \(\pi_2\) are the two projections of \(\mathrm{Hilb}_n(S) \times S\), and \(\gamma\) is contained in the subring \(R(S)\) of \(H^*(S)\) generated by the divisor classes.
The main theorem asserts that the cycle class map \(A^*(\mathrm{Hilb}_n(S)) \to H^*(\mathrm{Hilb}_n(S))\) is injective on the subring generated by small tautological classes for every \(K3\) surface \(S\) and positive integer \(n\). This verifies a weaker version of the Beauville-Voisin conjecture for the Hilbert scheme \(\mathrm{Hilb}_n(S)\). The idea in the proof of the theorem is to repackage the Chow rings in the language of representation theory by considering the action on \(A^*(\mathrm{Hilb}) = \bigoplus_{n = 0}^\infty A^*(\mathrm{Hilb}_n(S))\) of the Lie algebra \(\mathrm{Heis} \times \mathrm{Vir}\) where \(\mathrm{Heis}\) and \(\mathrm{Vir}\) are the Heisenberg algebra of I.~Grojnowski-H.~Nakajima and (the relative of) the Virasoro algebra of [\textit{M. Lehn}, Invent. Math. 136, No. 1, 157--207 (1999; Zbl 0919.14001)] respectively. The technical work is to lift from cohomology to Chow rings of the Virasoro algebra action of M.~Lehn and the \(W_{1+\infty}\) algebra action of \textit{W.-P. Li} et al. [Int. Math. Res. Not. 2002, No. 27, 1427--1456 (2002; Zbl 1062.14010)].
Section~2 recalls the basic materials about the Hilbert schemes of points on a \(K3\) surface and their Chow rings. Section~3 is devoted to the representation theory of Hilbert schemes, and verifies the analogue for \(A^*(\mathrm{Hilb})\) of the \(W_{1+\infty}\) algebra action of W.-P.~Li-Z.~Qin-W.~Wang. The main theorem is proved in Section~4. Section~5 deals with the representation theory of tautological classes on \(\mathrm{Hilb}_n(S)\). In Section~6, the authors investigate the geometry of nested Hilbert schemes and verifies the analogue for \(A^*(\mathrm{Hilb})\) of the Virasoro algebra action of M.~Lehn.
Reviewer: Zhenbo Qin (Columbia)Virtual cycles on projective completions and quantum Lefschetz formulahttps://zbmath.org/1491.140102022-09-13T20:28:31.338867Z"Oh, Jeongseok"https://zbmath.org/authors/?q=ai:oh.jeongseokSummary: For a compact quasi-smooth derived scheme \(M\) with \((- 1)\)-shifted cotangent bundle \(N\), there are at least two ways to localise the virtual cycle of \(N\) to \(M\) via torus and cosection localisations, introduced by \textit{Y. Jiang} and \textit{R. P. Thomas} [J. Algebr. Geom. 26, No. 2, 379--397 (2017; Zbl 1401.14221)]. We produce virtual cycles on both the projective completion \(\overline{N} : = \mathbb{P}(N \oplus \mathcal{O}_M)\) and projectivisation \(\mathbb{P}(N)\) and show the ones on \(\overline{N}\) push down to Jiang-Thomas cycles and the one on \(\mathbb{P}(N)\) computes the difference.
Using similar ideas we give an expression for the difference of the quintic and \(t\)-twisted quintic GW invariants of \textit{S. Guo}, \textit{F. Janda} and \textit{Y. Ruan} [``Structure of higher genus Gromov-Witten invariants of quintic 3-folds'', Preprint, \url{arXiv:1812.11908}].The direct image of generalized divisors and the norm map between compactified Jacobianshttps://zbmath.org/1491.140112022-09-13T20:28:31.338867Z"Carbone, Raffaele Marco"https://zbmath.org/authors/?q=ai:carbone.raffaele-marcoLet \(X\) be a curve, an embeddable noetherian scheme of pure dimension 1. First introduced by Hartshorne, the \textit{generalized divisors} are non-degenerate fractional ideals of \(\mathcal{O}_X\)-modules. Generalized divisors up to linear equivalence are equivalent to \textit{generalized line bundles}, i.e. pure coherent
sheaves which are locally free of rank 1 at each generic point. Further generalization is the notion of \textit{torsion-free sheaves of rank 1}.
Now, let \(\pi: X\rightarrow Y\) be a finite, flat morphism between noetherian curves. The goal of the paper under review is to define and study direct and inverse image for generalized divisors and for generalized line bundles on \( X \) and on \(Y\). Moreover, in the cases where \( X \) and \(Y\) are projective curves over a field (possibly
reducible, non-reduced) and the codomain curve is smooth, the author discusses the same notions for families of effective generalized divisors, parametrized by the Hilbert scheme. The same assumptions are required to introduce the notion of compactified Jacobians parametrizing torsion-free rank-1 sheaves and to study the Norm and the inverse image maps between them. Finally, the author consider the fibers of the Norm map and introduces the Prym stack as the fiber over the trivial sheaf.
Reviewer: Irene Spelta (Pavia)On the dimension of the global sections of the adjoint bundle for polarized 5-foldshttps://zbmath.org/1491.140122022-09-13T20:28:31.338867Z"Fukuma, Yoshiaki"https://zbmath.org/authors/?q=ai:fukuma.yoshiakiIn the study of complex polarized manifolds \((X,L)\) of dimension \(n\), an extensive literature is devoted to the properties of the adjoint bundles \(K_X+mL\) for some positive integer \(m\). In addition to nefness, point freeness and very ampleness, the positivity of the dimension \(h^0(K_X + m L)\) has been considered. A conjecture of Beltrametti and Sommese claiming that the nefness of \(K_X+(n-1)L\) implies its effectiveness attracted the attention of several researchers. In particular, Höring proved that this conjecture is true provided that \(h^0(L)>0\) [\textit{A. Höring}, J. Algebr. Geom. 21, No. 4, 721--751 (2012; Zbl 1253.14007)]. More recently, another conjecture concerning higher values of \(m\) was proposed by the author of the paper under review for \(n \geq 3\). It claims that \(h^0(K_X + m L) \geq \binom{m-1}{n}\) for every integer \(m \geq n + 1\) and whenever equality occurs for some \(m\) in this range, then \((X, L) = (\mathbb P^n, \mathcal O_{\mathbb P^n}(1))\) [\textit{Y. Fukuma}, Kyushu J. Math. 71, No. 1, 115--128 (2017; Zbl 1366.14010)].
In previous papers he succeeded to prove that this is true in some instances, in particular for \(n\leq 4\) and for \(n\geq 5\) and \(\dim(\mathrm{Bs}|L|) \leq 1\). In the paper under review the conjecture is proven for \(n=5\) and \(h^0(L)>0\). When \(K_X+4L\) is not nef the question is easily settled, while if \(K_X+4L\) is nef the proof is splitted into various cases depending on whether \(K_X+2L\) and \(K_X+L\) are effective or not.
Reviewer: Antonio Lanteri (Milano)Factorial nodal complete intersection 3-folds in \(\mathbb{P}^5\)https://zbmath.org/1491.140132022-09-13T20:28:31.338867Z"Hong, Kyusik"https://zbmath.org/authors/?q=ai:hong.kyusikSummary: Let \(X\) be a nodal complete intersection 3-fold defined by a hypersurface in \(\mathbb{P}^5\) of degree \(n\) and a smooth quadratic hypersurface in \(\mathbb{P}^5\). Then we show that \(X\) is factorial if it has at most \(n^2 -n+1\) nodes and contains no 2-planes, where \(n = 3, 4\).On the Hodge conjecture for quasi-smooth intersections in toric varietieshttps://zbmath.org/1491.140142022-09-13T20:28:31.338867Z"Bruzzo, Ugo"https://zbmath.org/authors/?q=ai:bruzzo.ugo"Montoya, William"https://zbmath.org/authors/?q=ai:montoya.williamThe Hodge conjecture for smooth hypersurfaces or smooth complete intersections in a projective space remains open. In general, one does not know whether the Hodge conjecture holds for a smooth hypersurface in the \textit{Noether-Lefschetz locus} [\textit{A. Dan} and \textit{I. Kaur}, C. R., Math., Acad. Sci. Paris 354, No. 3, 297--300 (2016; Zbl 1380.14001)]. In the paper under review, the authors proved two results. The first answers the Hodge conjecture for a very general quasi-smooth intersection in toric varieties. The second is about the Hodge conjecture for a quasi-smooth hypersurface in the Noether-Lefschetz locus.
Let \(\mathbb{P}^d_\Sigma\) be an Oda projective simplicial toric variety, and \(X\) be a very general quasi-smooth intersection cut off by \(f_1,\ldots, f_s\) such that the \(\dim X=d-s\) is even, and the divisor
\[
\sum^{s}_{i=1} \operatorname{div}(f_i)-\beta_0
\]
is nef. Then any Hodge class in the middle cohomology of \(X\) is algebraic, which is the restriction of an algebraic class on \(\mathbb{P}^d_\Sigma\).
Assume that the Picard number of the Oda variety \(\mathbb{P}^{2k+1}_\Sigma\) is one. Let \(X_f\) be a quasi-smooth hypersurface in \(\mathbb{P}^{2k+1}_\Sigma\), and \(\lambda\in H^{k, k}(X_f)_p\) be a non-trivial primitive Hodge class. With suitable condition on the local Nother-Lefschetz locus \(NL^k_\lambda\) with respect to the class \(\lambda\), the Hodge conjecture holds for any quasi-smooth hypersurface in \(NL^k_\lambda\).
Reviewer: Renjie Lyu (Beijing)On product identities and the Chow rings of holomorphic symplectic varietieshttps://zbmath.org/1491.140152022-09-13T20:28:31.338867Z"Barros, Ignacio"https://zbmath.org/authors/?q=ai:barros.ignacio"Flapan, Laure"https://zbmath.org/authors/?q=ai:flapan.laure"Marian, Alina"https://zbmath.org/authors/?q=ai:marian.alina"Silversmith, Rob"https://zbmath.org/authors/?q=ai:silversmith.robThe main purpose of the paper is to understand the Chow ring of irreducible holomorphic symplectic varieties. The authors conjecture a series of identities in the Chow rings \(\mathrm{CH}_{\star}(M \times X^{l})\), \(l \geq 1\), for a moduli space \(M\) of stable sheaves on a \(K3\) surface \(X\), inspired by the classical Beauville-Voisin identity for a \(K3\) surface. Assuming the conjectures are true all tautological classes fit in the lowest piece of a natural filtration emerging on \(\mathrm{CH}_{\star}(M)\).
In the two dimensional setting [\textit{A. Beauville} and \textit{C. Voisin}, J. Algebr. Geom. 13, No. 3, 417--426 (2004; Zbl 1069.14006)] an essential role in approaching the cycle structure is played by a distinguished zero-cycle \(c_X\). The cycle \(c_X\) has degree one and is the Chow class of any point lying on a rational curve in \(X\). The intersection of any two divisors is a multiple of \(c_X\), while the second Chern class of the tangent bundle satisfies \(c_2(T X) = 24c_X\).
In higher dimensions, they consider a moduli space \(M\) of stable sheaves on a \(K3\) surface with respect to a Mukai vector \(v\), which is a smooth projective irreducible holomorphic symplectic variety ([\textit{D. Huybrechts}, Lectures on \(K3\) surfaces. Cambridge: Cambridge University Press (2016; Zbl 1360.14099)]) of dimension \(v^{2}+2\). Also in this case there is a distinguished zero-cycle \(c_M \in \mathrm{CH}_0(M)\) of degree one: this is the class of any stable sheaf \(\mathcal{F}\) such that \(c_2(\mathcal{F})=k c_X\) in \(\mathrm{CH}_0(X)\), where \(k\) is the degree of the second Chern class specified by the Mukai vector \(v\).
As in the two dimensional case, one expects ([\textit{J. Shen} et al., Compos. Math. 156, No. 1, 179--197 (2020; Zbl 1436.14071),\textit{C. Voisin}, Prog. Math. 315, 365--399 (2016; Zbl 1352.32010)]) that the special cycle corresponds to the largest rational equivalence orbit of points on \(M\), but the intersection properties of \(c_M\) are not understood as well as those of the cycle in the two-dimensional context.
Fix the notation for the projections: \(\pi:M \times X \to M\) and \(\rho: M \times X \to X\). The authors consider a universal sheaf \(\mathcal{F} \to M \times X\) and they study its geometry and the geometry of the special cycles \(c_X\) and \(c_M\) in two steps: the first step describes the tautological subring as \(R_{\star}(X)=\mathrm{CH}_{2}(X) + \mathrm{CH}_{1}(X) + \mathbb{Z} c_{X}\subset \mathrm{CH}_{\star}(X)\), in the second step they focus on the rank zero virtual sheaf \(\overline{\mathcal{F}}=\mathcal{F}-\rho^{*}(\mathcal{F})\) with \(\mathcal{F}\in M\) such that \([\mathcal{F}]=c_{M}\in \mathrm{CH}_{0}(M)\).
These two steps bring to Conjecture 1 that states that for every \(i_1,\ldots,i_l \geq 0\), it holds that
\[
\alpha \cdot \ \mathrm{ch}_{i_1}(\overline{\mathcal{F}_1})\cdots \mathrm{ch}_{i_l}(\overline{\mathcal{F}_l})=0 \in \mathrm{CH}_{\star}(M\times X^l)
\]
where \( \alpha \in R_{\star}(M)\) is a tautological class of codimension \(d\), and \(\mathcal{F}_i\) denote the pullback to \(M\times X^l\) of the virtual universal sheaf on \(M\) and the i-th factor of \(X\).
Conjecture 1 brings to a generalization of identities that hold for \(K3\) surfaces, in a more general setup of moduli spaces. Moreover the identities of Conjecture 1 bring to a large collection of conjectural Chow vanishings in the self-products \(M \times M \times \cdots \times M\). They set \(\overline{\Delta}=\Delta - M\times c_M \in \mathrm{CH}_m(M \times M)\).
In Theorem 1 they show that the system of identities of Conjecture 1 is equivalent to the vanishing \(\alpha \cdot \ \overline{\Delta}_{0,1} \cdots \overline{\Delta}_{0,l}=0 \in \mathrm{CH}_{\star}(M \times M^{l})\), for any tautological class \(\alpha \in R_{\star}(M)\) of codimension \(d\) and integer \(l\) satisfying \(d + l > \dim M\). The theorem has many interesting consequences: the tautological ring has rank 1 in dimension 0, i.e. \(R_{\star}(M)=\mathbb{Q}\cdot c_M\), the modified diagonal cycle \(\Gamma^{m+1}(M,c_M)=\Delta - \Delta_{c} + \Delta_{c,c}- \cdots + \Delta_{c,c,\ldots,c}\) vanishes in \(\mathrm{CH}_{m}(M^{m+1})\).
Finally for every codimension \(0 \leq d \leq m\) they consider the increasing filtration
\[
S_{0} \subset S_1 \ldots \subset S_i \subset \ldots \subset \mathrm{CH}^{d}(M),
\]
where \(S_i(\mathrm{CH}^{d}(M))=\{ \alpha \ with \ \alpha \cdot \overline{\Delta}_{0,1} \ldots \overline{\Delta}_{0,m-d+i+1}=0 \subset \mathrm{CH}_{\star}(M\times M^{m-d+i+1})\}\). The filtration terminates [\textit{A. Beauville} and \textit{C. Voisin}, J. Algebr. Geom. 13, No. 3, 417--426 (2004; Zbl 1069.14006)] and the authors can prove that \(S_d(\mathrm{CH}^{d}(M))=\mathrm{CH}^{d}(M)\). Finally they remark that a stronger conjecture can be formulate, i.e Conjecture 2, that says that the restriction of the cycle class map to the tautological subring is injective. Anyway Conjecture 1 is easier to study since concerns in explicit relations in the Chow ring.
As evidence for Conjecture 1 they prove that it holds true for Hilbert schemes of \(n\) points on a smooth projective \(K3\) surface.
Reviewer: Annalisa Grossi (Chemnitz)A note on a Griffiths-type ring for complete intersections in Grassmannianshttps://zbmath.org/1491.140162022-09-13T20:28:31.338867Z"Fatighenti, Enrico"https://zbmath.org/authors/?q=ai:fatighenti.enrico"Mongardi, Giovanni"https://zbmath.org/authors/?q=ai:mongardi.giovanniLet \(X=\mathbf{V}(f)\subset\mathbb P^{n+1}\) be a smooth hypersurface of degree \(\deg f=d\). A classical result of [\textit{P. A. Griffiths}, Am. J. Math. 90, 805--865 (1968; Zbl 0183.25501)] identifies the primitive part of \(H^{n-p+1,p-1}(X)\) with the \(pd-n-2\) homogeneous component of \(\mathbb C[x_0,\ldots,x_{n+1}]/J_f\), where \(J_f=(f_0,\ldots,f_{n+1})\) is the Jacobian ideal generated by the partial derivatives of \(f\).
There exist various generalizations of this statement. For instance to complete intersections in projective space by [\textit{A. Dimca}, Duke Math. J. 78, No. 1, 89--100 (1995; Zbl 0839.14009)], or toric varieties by [\textit{V. V. Batyrev} and \textit{D. A. Cox}, Duke Math. J. 75, No. 2, 293--338 (1994; Zbl 0851.14021)], [\textit{K. Konno}, Compos. Math. 78, No. 3, 271--296 (1991; Zbl 0737.14002)], [\textit{A. R. Mavlyutov}, Pac. J. Math. 191, No. 1, 133--144 (1999; Zbl 1032.14013)], or hypersurfaces of high degree in arbitrary projective manifolds by [\textit{M. L. Green}, Compos. Math. 55, 135--156 (1985; Zbl 0588.14004)], or recently to zero loci of homogeneous vector bundles by [\textit{A. Huang} et al., ``Jacobian rings for homogenous vector bundles and applications'', Preprint, \url{arXiv:1801.08261}].
The main results of this paper are generalizations of Griffiths' result to complete intersections in Grassmann varieties. They do not use primitive cohomology, but a refinement, the vanishing cohomology. For hypersurfaces in projective space the distinction is irrelevant.
For instance if \(X\subset G=\mathrm{Gr}(k,n)\) is a smooth hypersurface of degree \(d\geq n-1\) and dimension \(N-1=k(n-k)-1\), and \(N\) is odd, then the vanishing part of \(H^{N-1-p,p}(X)\) is isomorphic to the degree \((p+1)d-n\) part of \(R^G_f\). The latter is what the authors call the Griffiths ring of \(X\). It is the quotient of the Plücker coordinate ring of \(G\) by the homogeneous ideal generated by \(f\) and its \(\mathfrak{sl}_n\)-orbit. Here \(\mathfrak{sl}_n\) acts by derivations in a concrete way. When \(N\) is even, then we may no longer obtain an isomorphism, but the difference is explicit, determined only by \(G\) and \(p\).
The authors prove a similar result for complete intersections of more than one hypersurface in \(G\). The Griffiths-type ring is now denoted \(\mathcal U\). Its definition is cohomological, motivated by a Cayley trick to reduce from complete intersections in \(G\) to hypersurfaces in a projective bundle over \(G\).
The main advantage of the results of this paper over other generalizations is their explicit nature, mirroring the original result of Griffiths. The authors give concrete presentations of their Griffiths-type rings and use them to compute several explicit examples of Hodge groups. For instance they carry out these computations for Fano 5-folds and 4-folds of genus 6 and degree 10, a Calabi-Yau section of \(\mathrm{Gr}(2,7)\), and for Fano varieties of K3-type. Some of these computations of Hodge groups were known, but the ring structure is new.
Apart from classical methods such as chasing exact sequences in cohomology, the paper makes great use of the Cayley trick for reducing complete intersections to hypersurfaces in a projective bundle, and of the connection between the Hodge theory of a particular class of projectively normal \(X\) and the infinitesimal first-order deformation module of the affine cone \(A_X\).
Reviewer: Mihai Fulger (Storrs)Complete families of indecomposable non-simple abelian varietieshttps://zbmath.org/1491.140172022-09-13T20:28:31.338867Z"Flapan, Laure"https://zbmath.org/authors/?q=ai:flapan.laureThe main result of the paper shows how to construct complete families of indecomposable abelian varieties whose very general fiber is isogenous to a prescribed one and whose monodromy group is either a product of symplectic groups or a unitary group. The result applies to the study of the monodromy group of families of abelian varieties: it shows how to realize any product of symplectic groups of total rank g as the connected monodromy group of a complete family of $g'$-dimensional abelian varieties for any $g'\geq g$. This has a direct relation with the study of variations of Hodge structures on families of non-simple Jacobians, for instance Jacobians of coverings of curves. As a further application, it allows to recover certain Kodaira fibrations precedently found by the author with different methods and to construct a new one with fibers of genus 4.
Reviewer: Sara Torelli (Pavia)On the monodromy invariant Hermitian form for \(A\)-hypergeometric systemshttps://zbmath.org/1491.140182022-09-13T20:28:31.338867Z"Verschoor, Carlo"https://zbmath.org/authors/?q=ai:verschoor.carlo\(A\)-hypergeometric functions are a generalization (introduced by \textit{I. M. Gelfand} et al. [Adv. Math. 84, No. 2, 255--271 (1990; Zbl 0741.33011)]) of hypergeometric functions. An \(A\)-hypergeometric system comprises Euler operators and box operators, defined by means of a lattice in \(\mathbb{Z}^N\) satisfying certain conditions. \textit{F. Beukers} [J. Reine Angew. Math. 718, 183--206 (2016; Zbl 1355.33017)] has shown how to find (under very restrictive conditions) a subgroup of the full monodromy group using Mellin-Barnes integral solutions of the associated \(A\)-hypergeometric system. In the present paper an explicit construction of the invariant Hermitian form for the monodromy of an \(A\)-hypergeometric system is given provided that there is a Mellin-Barnes basis of solutions.
Reviewer: Vladimir P. Kostov (Nice)Perverse Leray filtration and specialisation with applications to the Hitchin morphismhttps://zbmath.org/1491.140192022-09-13T20:28:31.338867Z"De Cataldo, Mark Andrea A."https://zbmath.org/authors/?q=ai:de-cataldo.mark-andrea-aThe author develops a general framework to study the specialisation morphism as a filtered morphism for perverse Leray filtration. In fact, some results have been obtained in the context of the Hitchin morphism associated with a family of smooth varieties over a base [\textit{L. Migliorini} and \textit{V. Shende}, Algebr. Geom. 5, No. 1, 114--130 (2018; Zbl 1406.14005)].
The paper is organized as follows: In Section 1, the author lists the notations and introduces the formalism of the vanishing functors and nearby cycles functors. Section 2 is devoted to discuss the morphisms of type \(\delta\), which are used to measure the failure of the restriction-to-special-fiber functor to commute with perverse truncation. In Section 3, the author develops the aforementioned framework. Section 4 applies the abstract framework in the case of the Hitchin morphisms arising from the compactification of Dolbeault moduli spaces introduced in [\textit{M. A. A. de Cataldo}, Int. Math. Res. Not. 2021, No. 5, 3543--3570 (2021; Zbl 1474.14073)].
Reviewer: Yonghong Huang (Guangzhou)Stable reflexive sheaves and localizationhttps://zbmath.org/1491.140202022-09-13T20:28:31.338867Z"Gholampour, Amin"https://zbmath.org/authors/?q=ai:gholampour.amin"Kool, Martijn"https://zbmath.org/authors/?q=ai:kool.martijnThe paper studies the moduli space \(\mathcal{N}(2,c_1,c_2,c_3)\) of rank 2 \(\mu\)-stable reflexive sheaves on \(\mathbb{P}^3\). The action of the torus \(T\) on \(\mathbb{P}^3\) lifts to \(\mathcal{N}(2,c_1,c_2,c_3)\). With the first and second Chern classes fixed, the authors compute the generating function \(Z_{c_1,c_2}^{refl}(q):=\sum_{c_3}e(\mathcal{N}(2,c_1,c_2,c_3))q^{c_3}\) by counting the \(T\)-fixed points in \(\mathcal{N}(2,c_1,c_2,c_3)\).
Every \(T\)-equivariant rank 2 reflexive sheaf on \(\mathbb{P}^3\) can be described by its toric data \[(\mathbf{u},\mathbf{v},\mathbf{p}):=\{(u_i,v_i,p_i)\}_{i=1,2,3,4},\] where \(u_i\in\mathbb{Z},v_i\in\mathbb{Z}_{\geq 0}\) and \(p_i\in Gr(1,2)\cong\mathbb{P}^2\).
Every \(T\)-fixed sheaf has a unique \(T\)-equivariant structure with \(u_1=u_2=u_3=0\) in the toric data. Therefore the number of \(T\)-fixed points in \(\mathcal{N}(2,c_1,c_2,c_3)\) can be obtained by counting the toric data with extra condition \(u_1=u_2=u_3=0\) which turns out to be a combinatorial problem.
The authors classify \(T\)-equivariant rank 2 reflexive sheaves into three types by their toric data. The final formula of \(Z_{c_1,c_2}^{refl}(q)\) is given by counting sheaves of each type and adding the numbers together.
Another result of the paper is that the third Chern class \(c_3\) of any rank 2 reflexive sheaf must be non-negative and must have an upper bound given by \(c_1\) and \(c_2\). Moreover, the authors reprove the Hartshorne inequality for \(T\)-equivariant cases, which gives explicit upper bounds for \(c_3\). Those results imply that \(Z_{c_1,c_2}^{refl}(q)\) is a polynomial.
Let \(\mathcal{M}(2,c_1,c_2,c_3)\) be the moduli space of rank 2 \(\mu\)-stable sheaves. If one wants to deduce the generating function \(Z_{c_1}(p,q):=\sum_{c_2,c_3}e(\mathcal{M}(2,c_1,c_2,c_3))p^{c_2}q^{c_3}\) from \(Z_{c_1}^{refl}(p,q):=\sum_{c_2}Z_{c_1,c_2}^{refl}(q)p^{c_2}\), one need study the Euler numbers of some Quot-schemes. The paper deals with the case \(c_1=-1,c_2=1\) and gives a closed formula. For a general case, it could be very complicated.
Finally, the authors discuss briefly the generalization of the technics of this paper to any smooth projective toric 3-fold. Also, they look at the wall-crossing phenomena. But this is a short chapter and no concrete result is provided.
Reviewer: Yao Yuan (Beijing)On the rationality of moduli spaces of vector bundles over chain-like curveshttps://zbmath.org/1491.140212022-09-13T20:28:31.338867Z"Suhas, B. N."https://zbmath.org/authors/?q=ai:suhas.b-n"Roy, Praveen Kumar"https://zbmath.org/authors/?q=ai:roy.praveen-kumar"Singh, Amit Kumar"https://zbmath.org/authors/?q=ai:singh.amit-kumar.1Summary: Let \(C\) be a chain-like curve over \(\mathbb{C} \). In this paper, we investigate the rationality of moduli spaces of \(w\)-semistable vector bundles on \(C\) of arbitrary rank and fixed determinant by putting some restrictions on the Euler characteristics.The dual complex of a semi-log canonical surfacehttps://zbmath.org/1491.140222022-09-13T20:28:31.338867Z"Brown, Morgan V."https://zbmath.org/authors/?q=ai:brown.morgan-vSummary: Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary \(\Delta\) of a log canonical pair \((X,\Delta)\) and also appear as limits of canonically polarized varieties in moduli theory. For certain three-fold pairs \((X,\Delta)\), we show how to compute the PL homeomorphism type of the dual complex of a dlt minimal model directly from the normalization data of \(\Delta\).Birational geometry of blow-ups of projective spaces along points and lineshttps://zbmath.org/1491.140232022-09-13T20:28:31.338867Z"He, Zhuang"https://zbmath.org/authors/?q=ai:he.zhuang"Yang, Lei"https://zbmath.org/authors/?q=ai:yang.lei.2Let \(p_0,\ldots, p_5\) six points in (very) general position in \({\mathbb P}_{\mathbb C}^3\). Then:
\begin{itemize}
\item Let \(u: Y\to {\mathbb P}_{\mathbb C}^3\) be the succesive blow up at \(p_0,\ldots, p_5\) and the proper transforms of the \(9\) lines \(\overline{p_ip_j}\) labeled by \[(ij)\in {\mathcal I}=\{03,04,34, 12, 15, 25, 05, 13, 24\};\]
\item Let \(v: X \to {\mathbb P}_{\mathbb C}^3\) be the succesive blow up at \(p_0,\ldots, p_5\) and the proper transforms of all the 15 lines \(\overline{p_ip_j}\).
\end{itemize}
Let \(E_i\) and \(E_{ij}\) be the exceptional divisors of the blow-ups \(Y\) and \(X\) over the points \(p_i\) and the lines \(\overline{p_ip_j}\). Let \(H:=u^*{\mathcal O}_{{\mathbb P}_{\mathbb C}^3}(1)\). Then the Picard group of \(Y\) is freely generated by \(H\), \(E_i\) and \(E_{ij}\). Consider the class over \(Y\) and \(X\), \[D:= 13H-7(E_i+E_2+E_5)-5(E_0+E_3+E_4)-\] \[-3(E_{03}+E_{04}+E_{34})-4(E_{05}+E_{13}+E_{24})-(E_{12}+E_{15}+E_{25}).\]
With the previous notation, the following results are proved.
Theorem 1. For very general six points as above:
\begin{enumerate}
\item The lineal system \(|D|\) has dimension 3, hence it determines a map \(\phi_D: Y--->{\mathbb P}_{\mathbb C}^3\).
\item There exist 6 points \(q_0,\ldots,q_5\) in the target copy of \({\mathbb P}_{{\mathbb C}^3}\) that are projectively equivalent to \(p_0,\ldots,p_5\). Blowing up the six points and the corresponding 9 lines \(\overline{q_iq_j}\) for \((ij)\in {\mathcal I}\) induces a pseudo-automorphism \(\phi: Y ---> Y\) (i.e. there are open subsets \(U\) and \(V\) in \(Y\) of codimension at least 2, such that \(\phi\) induces an isomorphism \(\phi: U\to V\)). Blowing up \(q_i\) and all 15 lines \(\overline{q_iq_j}\) induces a pseudo-automorphism \(\phi_X: X ---> X\).
\item The pseudo-automorphisms \(\phi\) and \(\phi_X\) are of infinite order.
\item The effective cone \(\overline{\text{Eff}}(Y)\) of \(Y\) has infinitely many extremal rays. Hence, \(\overline{\text{Eff}}(Y)\) is not rational polyhedral and \(Y\) is not a Mori Dream Space. The same results hold for \(X\).
\end{enumerate}
Theorem 2. For general six points as above:
\begin{enumerate}
\item \(X\) has a unique anticanonical section \(S\), which is a Jacobian \(K3\) Kummer surface with Picard rank \(\rho(S)=17\).
\item The restriction of \(\phi_X\) to \(S\) equals Keum's authomorphism \(\kappa: S\to S\) associated the Weber Hexad \({\mathcal H}=\{1,2,5,12,14,23\}\).
\item The inverse \(\phi_X^{-1}\) (and \(\phi^{-1}\)) is induced by a complete linear system of \(D'\) where: \[D':=13H-5(E_1+E_2+E_5)-7(E_0+E_3+E_4)+\] \[-(E_{03}+E_{04}+E_{34})-4(E_{05}+E_{13}+E_{24})-3(E_{12}+E_{15}+E_{25}).\]
\end{enumerate}
Now let \(\psi: {\mathbb P}_{\mathbb C}^3 ---> {\mathbb P}_{\mathbb C}^3\) be the birational automorphism induced by \(|D|\). Let \(H_n\subset \text{Bir}({\mathbb P}_{\mathbb C}^n)\) be the subset of the group of birational automorphisms of \({\mathbb P}_{\mathbb C}^n\) that consists on those automorphism that only contract rational hypersurfaces. Then \(G_n\subset H_n\), where \(G_n:=\langle \text{PGL}(n+1), \sigma_n\rangle\) and \(\sigma_n\) is the standard Cremona transformation of \({\mathbb P}_{\mathbb C}^n\).
Theorem 3. With the same notation as above, \(\psi\in H_3\) but \(\psi \notin G_3\).
For \(n\geq 3\) let \(Y_n\) be the blow-up of \({\mathbb P}_{\mathbb C}^n\) at \(n+3\) points in very general position and 9 lines through six of them such that when the six points are indexed by 0 to 5, the 9 lines are labeled by \({\mathcal I}\).
A variety \(X\) is \({\mathbb Q}\)-factorial if for any Weil divisor \(D\) on X, there exists some integer \(m\) such that \(mD\) is Cartier. A small \({\mathbb Q}\)-factorial modification (SQM) of \(X\) is a rational map \(g : X --- > X'\) such that \(X'\) is \({\mathbb Q}\)-factorial and \(g\) is an isomorphism in codimension 1.
Theorem 4. For each \(n\geq 4\) there is a small \({\mathbb Q}\)-factorial modification (SQM) \(\tilde{Y}_n\) of \(Y_n\) such that \(\tilde{Y}_n\) is a \({\mathbb P}_{\mathbb C}^1\)-bundle over \(Y_{n-1}\). For \(n\geq 3\), \(\overline{\text{Eff}}(Y_n)\) has infinitely many extremal rays. Hence, \(Y_n\) are not Mori Dream for \(n\geq 3\).
Denoting by \(\overline{M}_{g,n}\) the Deligne-Mumford compactification of the moduly of space of stable curves of genus \(g\) with \(n\) marked points, the following is proved:
Theorem 5. For \(n\geq 7\), the effective cone of the blow-up of \(\overline{M}_{0,n}\) at a very general point has infinitely many extremal rays. Hence, the blow-up of \(\overline{M}_{0,n}\) at a very general point is not a Mori Dream Space.
Reviewer: Ana Bravo (Madrid)Birational geometry for the covering of a nilpotent orbit closure. II.https://zbmath.org/1491.140242022-09-13T20:28:31.338867Z"Namikawa, Yoshinori"https://zbmath.org/authors/?q=ai:namikawa.yoshinoriLet \(O\) be a nilpotent orbit of a complex semisimple Lie algebra \(\mathfrak{g}\) and let \(\pi: X \to \bar{O}\) be the finite covering associated with the universal covering of \(O\). In [\textit{Y. Namikawa}, in: Handbook of moduli. Volume III. Somerville, MA: International Press; Beijing: Higher Education Press. 1--38 (2013; Zbl 1322.14013)], he has explicitly constructed a \(\mathbb{Q}\)-factorial terminalization \(\widetilde{X}\) of \(X\) when \(\mathfrak{g}\) is classical.
In this paper, the author counts different \(\mathbb{Q}\)-factorial terminalizations of \(X\). He also constructs the universal Poisson deformation of \(\widetilde{X}\) over \(H^2(\widetilde{X}, \mathbb{C})\) and looks at the action of the Weyl group \(W(X)\) on \(H^2(\widetilde{X}, \mathbb{C})\), leading toward an explicit geometric description of \(W(X)\).
Reviewer: Mee Seong Im (Annapolis)The wild McKay correspondence for cyclic groups of prime power orderhttps://zbmath.org/1491.140252022-09-13T20:28:31.338867Z"Tanno, Mahito"https://zbmath.org/authors/?q=ai:tanno.mahito"Yasuda, Takehiko"https://zbmath.org/authors/?q=ai:yasuda.takehikoSummary: The \(v\)-function is a key ingredient in the wild McKay correspondence. In this paper, we give a formula to compute it in terms of valuations of Witt vectors, when the given group is a cyclic group of prime power order. We apply it to study singularities of a quotient variety by a cyclic group of prime square order. We give a criterion whether the stringy motive of the quotient variety converges or not. Furthermore, if the given representation is indecomposable, then we also give a simple criterion for the quotient variety being terminal, canonical, log canonical, and not log canonical. With this criterion, we obtain more examples of quotient varieties which are Kawamata log terminal (klt) but not Cohen-Macaulay.Cyclic branched coverings of surfaces with abelian quotient singularitieshttps://zbmath.org/1491.140262022-09-13T20:28:31.338867Z"Bartolo, E. Artal"https://zbmath.org/authors/?q=ai:artal-bartolo.enrique"Cogolludo-Agustin, J. I."https://zbmath.org/authors/?q=ai:cogolludo-agustin.jose-ignacio"Martin-Morales, J."https://zbmath.org/authors/?q=ai:martin-morales.jorgeIn the paper under review, the authors study the problem of describing the irregularity of cyclic branched coverings of surfaces with abelian quotient singularities, and they use this description to find formulas for the particular case of the weighted projective plane \(\mathbb{P}^{2}_{w}\). Let us recall that an abelian quotient singular point in dimension two is necessarily a cyclic singularity. The main result of this paper describes the dimension of the equivariant spaces of the first cohomology of a \(d\)-cyclic cover \(\rho : X \rightarrow \mathbb{P}^{2}_{w}\) ramified along a curve \[C = \sum_{j}n_{j}C_{j}.\] The cover \(\rho\) naturally defines a divisor \(H\) such that \(dH\) is linearly equivalent to \(C\). If \(K_{\mathbb{P}^{2}_{w}}\) is the canonical divisor of \(\mathbb{P}^{2}_{w}\), and \[C^{(k)} = \sum_{j=1}^{r} \bigg\lfloor \frac{kn_{j}}{d}\bigg\rfloor C_{j}, \quad 0 \leq k < d,\] then these dimensions are given as the cokernel of the evaluation linear maps \[\pi^{(k)} : H^{0}(\mathbb{P}^{2}_{w}, \mathcal{O}_{\mathbb{P}^{2}_{w}}(kH + K_{\mathbb{P}^{2}_{w}} - C^{(k)})) \rightarrow \bigoplus_{P\in S}\frac{ \mathcal{O}_{\mathbb{P}^{2}_{w}}(kH + K_{\mathbb{P}^{2}_{w}} - C^{(k)})}{\mathcal{M}^{(k)}_{C,P}},\] where \(\mathcal{M}^{(k)}_{C,P}\) is defined as the quasi-adjunction-type \(\mathcal{O}_{\mathbb{P}^{2}_{w},P}\)-module, namely \[ \mathcal{M}^{(k)}_{C,P}:= \bigg\{ g \in \mathcal{O}_{\mathbb{P}^{2}_{w},P}(kH + K_{\mathbb{P}^{2}_{w}} - C^{(k)})\, : \, \mathrm{mult}_{E_{v}} \, \pi^{*}g > \sum_{j=1}^{r} \bigg[\frac{kn_{j}}{d}\bigg]m_{v_{j}}-v_{v}, \,\,\, \forall v \in \Gamma_{P} \bigg\}.\] Here the symbol \([\cdot ]\) denotes the decimal part of a rational number, and the multiplicities \(m_{v_{j}}\) and \(v_{v}\) are provided by \(\pi^{*}C_{j} = \hat{C}_{j} + \sum_{P \in S}\sum_{v \in \Gamma_{P}}m_{v_{j}}E_{v}\) and \(K_{\pi} = \sum_{P\in S}\sum_{v \in \Gamma_{P}}(v_{v}-1)E_{v}\) for an embedded \(\mathbb{Q}\)-resolution \(\pi\) of \(C \subset \mathbb{P}^{2}_{w}\), where \(S\) is the set of points of \(\mathbb{P}^{2}_{w}\) blown up in the resolution \(\pi\), and \(\Gamma_{P}\) is the dual graph associated with the resolution of \(P \in S\). As a consequence, one has \[h^{1}(X,\mathbb{C}) = 2\sum_{k=0}^{d-1}\dim\, \mathrm{coker} \, \pi^{(k)}.\]
As a non-trivial application of their studies, the authors present a Zariski pair of irreducible curves in a weighted projective plane, that is, two curves in the same plane \(\mathbb{P}^{2}_{w}\) with the same degree and local type of singularities, but whose embeddings are not homeomorphic.
Reviewer: Piotr Pokora (Kraków)On threefold canonical thresholdshttps://zbmath.org/1491.140272022-09-13T20:28:31.338867Z"Chen, Jheng-Jie"https://zbmath.org/authors/?q=ai:chen.jheng-jieSummary: We show that the set of threefold canonical thresholds satisfies the ascending chain condition. Moreover, we derive that threefold canonical thresholds in the interval \((\frac{ 1}{ 2}, 1)\) consists of \(\{\frac{ 1}{ 2} + \frac{ 1}{ n}\}_{n \geq 3} \cup \{\frac{ 4}{ 5}\}\).Numerical characterization of some toric fiber bundleshttps://zbmath.org/1491.140282022-09-13T20:28:31.338867Z"Druel, Stéphane"https://zbmath.org/authors/?q=ai:druel.stephane"Lo Bianco, Federico"https://zbmath.org/authors/?q=ai:lo-bianco.federicoLet \(X\) be a complex projective manifold with numerically flat tangent bundle, i.e. the tangent bundle \(T_X\) is nef and has trivial determinant. Then one obtains, as a consequence of Yau's theorem, that \(X\) is an étale quotient of an abelian variety. Let now \((X, D)\) be a log smooth projective pair, i.e. \(X\) is a projective manifold and \(D \subset X\) is a reduced divisor with simple normal crossings. If the logarithmic tangent bundle \(T_X(-\log D)\) is trivial a theorem of \textit{J. Winkelmann} [Osaka J. Math. 41, No. 2, 473--484 (2004; Zbl 1058.32011)] gives a complete classification, the simplest example being smooth toric varieties \(X\) with boundary divisor \(D\). The first result of this paper is a generalization of Winkelmann's result: if the logarithmic tangent bundle \(T_X(-\log D)\) is numerically flat, the manifold \(X\) is an analytic fibre bundle \(X \rightarrow T\) over a torus \(T\) such that the fibre \((F, D|_F)\) is a smooth toric variety. This result is obtained as a corollary of a much more general statement: let \(X\) be a normal projective variety of klt type, and let \(D\) be a reduced effective divisor such that the pair \((X, D)\) is log-canonical. Assume that the following holds:
\begin{itemize}
\item the divisor \(-(K_X+D)\) is nef;
\item the logarithmic cotangent sheaf \(\Omega^{[1]}_X(\log D)\) is locally free and numerically flat on every rational curve.
\end{itemize}
Then \(X\) is an analytic fibre bundle \(X \rightarrow T\) onto a manifold with \(K_T \equiv 0\) such that the fibre \((F, D|_F)\) is a toric variety with boundary divisor \(D|_F\). Moreover the manifold \(T\) does not contain any rational curves. Hence one expects that \(T\) is an étale quotient of an abelian variety, but this is a very difficult open problem if \(\dim T \geq 3\). Note that structure theorems for varieties with nef anticanonical bundle have been established recently [\textit{J. Cao} and \textit{A. Höring}, J. Algebr. Geom. 28, No. 3, 567--597 (2019; Zbl 1419.14055); \textit{F. Campana} et al., Algebr. Geom. 8, No. 4, 430--464 (2021; Zbl 1483.14016)], but they do not apply to pairs with reduced boundary divisor. The key point of the proof is to show that for some well-chosen birational model the foliation defined by the MRC-fibration is regular, since this implies that the MRC-fibration is holomorphic and in fact a submersion. An interesting technical statement involved in the proof is a descent lemma for vector bundles if the morphism has rationally chain-connected fibres, cf. also [\textit{I. Biswas} and \textit{J. P. P. dos Santos}, C. R., Math., Acad. Sci. Paris 347, No. 19--20, 1173--1176 (2009; Zbl 1175.14031); \textit{D. Greb} et al., Compos. Math. 155, No. 2, 289--323 (2019; Zbl 1443.14009)].
Reviewer: Andreas Höring (Nice)On boundedness of semistable sheaveshttps://zbmath.org/1491.140292022-09-13T20:28:31.338867Z"Langer, Adrian"https://zbmath.org/authors/?q=ai:langer.adrianRoughly speaking, a moduli space is a geometric space whose points correspond to isomorphism classes of objects we are interested in. It is of no doubt that the importance of moduli spaces in algebraic geometry. However, when we collect all the algebro-geometric objects (for instance, vector bundles on a given projective variety with fixed numerical invariants), we do not expect the existence of a nice moduli space. Hence, it is necessary to introduce the notion of stability, and try to construct moduli spaces of stable objects. To construct a projective moduli space as a result, the boundedness of the family of isomorphism classes of (semi-)stable objects is crucial. Therefore, the boundedness results of the family of isomorphism classes of \(H\)-semistable (\(H\) is an ample divisor on a smooth projective variety \(X\) over an algebraically closed field \(k\)) torsion-free coherent sheaves on \(X\) with fixed numerical invariants are foundational (cf. [\textit{M. Maruyama}, Moduli spaces of stable sheaves on schemes: restriction theorems, boundedness and the GIT construction. With collaboration of T. Abe and M. Inaba. Tokyo: Mathematical Society of Japan (2016; Zbl 1357.14017)] for \(\mathrm{char}(k)=0\), and [\textit{A. Langer}, Ann. Math. (2) 159, No. 1, 251--276 (2004; Zbl 1080.14014)] for \(\mathrm{char}(k)>0\)).
The paper under review provides a new simple proof of [\textit{A. Langer}, Ann. Math. (2) 159, No. 1, 251--276 (2004; Zbl 1080.14014)] without using characteristic \(p\) methods. Instead, the author reduces the statement to the case of projective spaces, and proves Bogomolov's inequality on \(\mathbb{P}^n\) without using restriction theorem (of Flenner, or Mehta-Ramanathan). The latter one provides a restriction theorem which implies the boundedness of the families over projective spaces. This new method works both for the characteristic zero case and the characteristic \(p>0\) case.
In Section 4, the author also provides an alternative proof of Bogomolov's inequality on smooth projective surfaces in the characteristic zero case without reducing to the curve case. Instead, the author uses a bound of the slope of the Frobenius pull-back and a bound the number of global sections.
Reviewer: Yeongrak Kim (Saarbrücken)Bernstein-Sato polynomials for general ideals vs. principal idealshttps://zbmath.org/1491.140302022-09-13T20:28:31.338867Z"Mustaţă, Mircea"https://zbmath.org/authors/?q=ai:mustata.mirceaSummary: We show that given an ideal \(\mathfrak{a}\) generated by regular functions \(f_1,\dots,f_r\) on \(X\), the Bernstein-Sato polynomial of \(\mathfrak{a}\) is equal to the reduced Bernstein-Sato polynomial of the function \(g=\sum_{i=1}^rf_iy_i\) on \(X\times\mathbb{A}^r\). By combining this with results from [\textit{N. Budur} et al., Compos. Math. 142, No. 3, 779--797 (2006; Zbl 1112.32014)], we relate invariants and properties of \(\mathfrak{a}\) to those of \(g\). We also use the result on Bernstein-Sato polynomials to show that the Strong Monodromy Conjecture for Igusa zeta functions of principal ideals implies a similar statement for arbitrary ideals.A remark on the Castelnuovo-Mumford regularity of powers of ideal sheaveshttps://zbmath.org/1491.140312022-09-13T20:28:31.338867Z"Shang, Shijie"https://zbmath.org/authors/?q=ai:shang.shijie|shang.shijie.1Summary: We show that a bound of Castelnuovo-Mumford regularity of any power of the ideal sheaf of a smooth projective complex variety \(X \subseteq \mathbb{P}^r\) is sharp exactly for complete intersections, provided the variety \(X\) is cut out scheme-theoretically by several hypersurfaces in \(\mathbb{P}^r\). This generalizes a result of \textit{A. Bertram} et al. [J. Am. Math. Soc. 4, No. 3, 587--602 (1991; Zbl 0762.14012)].Rigid connections and \(F\)-isocrystalshttps://zbmath.org/1491.140322022-09-13T20:28:31.338867Z"Esnault, Hélène"https://zbmath.org/authors/?q=ai:esnault.helene"Groechenig, Michael"https://zbmath.org/authors/?q=ai:groechenig.michaelThe primary objects to study in the paper are irreducible \emph{rigid} rank~\(r\) flat connections \((E, \nabla)\) with a \emph{torsion} determinant line bundle \(L\), over a smooth projective variety \(X/\mathbb{C}\). \textit{C. T. Simpson} [Publ. Math., Inst. Hautes Étud. Sci. 79, 47--129 (1994; Zbl 0891.14005)] conjectured that such connections are \emph{of geometric origin}, meaning that they are subquotients of Gauß-Manin connections of a family of smooth projective varieties defined on an open dense subvariety of \(X\). Moreover, it's known that Gauß-Manin connections in characteristic \(p\) have nilpotent \(p\)-curvatures. So, Simpson's conjecture predicts that mod-\(p\) reductions of rigid connections have nilpotent \(p\)-curvatures. The first main result of this paper confirms this expectation for \(p\) sufficiently large.
Based on the first result, the authors constructed \(F\)-isocrystalline realizations of such rigid connections. That is, they showed that there is a finite type \(\mathbb{Z}\)-scheme \(S\) over which \((X, (E, \nabla))\) has a model \((X_S, (E_S, \nabla_S))\), such that for all \(W(k)\)-points of \(S\), where \(k\) is a finite field, the \(p\)-adic completion of the base change \((\widehat{E}_{W(k)}, \widehat{\nabla}_{W(k)}))\) on \(\widehat{X}_{W(k)}\) defines a crystal on \(X_k/W(k)\) and the isocrystal \((\widehat{E}_{W(k)}, \widehat{\nabla}_{W(k)})) \otimes \mathbb{Q}\) is indeed an \(F\)-isocrystal after a base change to a finite field extension. There is a crystalline representation associated the constructed \(F\)-isocrystal, whose properties were also studied.
Proofs for those two main results rely on choices of arithmetic models \((X_S/S, L_S)\) of \((X/\mathbb{C}, L)\) (Lemma~3.1, Prop.~3.3, Prop.~4.10), and the fact that there are only finite number of isomorphism classes of rigid connections for fixed \(r\) and \(L\). An indispensable tool used in the proof of the second main theorem is the Higgs-de~Rham flow developed in [\textit{G. Lan} et al., J. Eur. Math. Soc. (JEMS) 21, No. 10, 3053--3112 (2019; Zbl 1444.14048)].
Finally, they showed that if there is an \(S\)-model \((X_S, (E_S, \nabla_S))\) of \((X, (E,\nabla))\) for a finite type \(\mathbb{Z}\)-scheme \(S\), satisfying that for each closed point \(s \in S\), \((E_s, \nabla_s)\) has \emph{vanishing} \(p\)-curvature, then \((E, \nabla)\) has \emph{unitary monodromy}. The proof is based on the study of the representation previously mentioned. This result can be seen as one step towards the understanding of the Grothendieck-Katz \(p\)-curvature conjecture.
A closely related paper to the one under review is~[\textit{H. Esnault} and \textit{M. Groechenig}, Sel. Math., New Ser. 24, No. 5, 4279--4292 (2018; Zbl 1408.14037)], where another conjecture from [\textit{C. T. Simpson}, Publ. Math., Inst. Hautes Étud. Sci. 79, 47--129 (1994; Zbl 0891.14005)] was proved. Indeed, the result there was originally proved in a preprint version of the current one, proved as a consequence of the second main result here, together with the theory of \(p\)-to-\(\ell\)-companions [\textit{T. Abe} and \textit{H. Esnault}, Ann. Sci. Éc. Norm. Supér. (4) 52, No. 5, 1243--1264 (2019; Zbl 1440.14097)]. In~[\textit{H. Esnault} and \textit{M. Groechenig}, Sel. Math., New Ser. 24, No. 5, 4279--4292 (2018; Zbl 1408.14037)], a shorter proof was given. Moreover, there the result was proved for \emph{quasi-project} varieties.
There are still questions left open, e.g., if the first result holds for small primes \(p\).
Reviewer: Yun Hao (Berlin)Tamely ramified torsors and parabolic bundleshttps://zbmath.org/1491.140332022-09-13T20:28:31.338867Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Borne, Niels"https://zbmath.org/authors/?q=ai:borne.nielsSummary: Given a variety \(X\), a normal crossings divisor \(D \subset X\), and a finite Abelian group scheme \(G\), we relate, in the case of abelian monodromy, the following two:
\begin{itemize}
\item existence of a \(G\)-torsor with prescribed ramification;
\item existence of essentially finite parabolic vector bundles with prescribed weights.
\end{itemize}Framed motives of algebraic varieties (after V. Voevodsky)https://zbmath.org/1491.140342022-09-13T20:28:31.338867Z"Garkusha, Grigory"https://zbmath.org/authors/?q=ai:garkusha.grigory"Panin, Ivan"https://zbmath.org/authors/?q=ai:panin.ivanIn this groundbreaking paper, the authors lay the foundation for motivic infinite loop space theory. Developing ideas of V.~Voevodsky (outlined in unpublished notes), they define framed correspondences and prove that they provide computationally accessible models for infinite loop spaces of suspension spectra of smooth schemes.
What does this mean? Motivic homotopy theory is a homotopy theory of smooth schemes introduced by \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)]. It has a stable variant in which smash product with the projective line \(\mathbb{P}^1\) becomes invertible. This forms a category of motivic spectra which is related to motivic spaces in the same way that topological spectra (the objects of classical stable homotopy theory) are related to topological spaces. In particular, we have an adjunction \((\Sigma^\infty_{\mathbb{P}^1},\Omega^{\infty}_{\mathbb{P}^1})\) between pointed motivic spaces and motivic spectra. As such, understanding the (nonnegative) homotopy groups (or sheaves) of \(\Omega^\infty_{\mathbb P^1}E\) is equivalent to understanding the (nonnegative, stable) homotopy groups (or sheaves) of \(E\).
Motivic spectra represent cohomology theories of algebro-geometric interest -- motivic cohomology, (homotopy) algebraic \(K\)-theory, Hermitian \(K\)-theory, and algebraic cobordism, to name a few -- and thus it is very desirable to have good models for motivic infinite loop spaces. The authors verify that framed correspondences provide such models for \(\mathbb P^1\)-suspension spectra of simplicial smooth schemes under mild hypotheses. Indeed, a mild reinterpretation of Theorem 1.3 (see also Theorem 10.7) says that for a smooth scheme \(X\) over an infinite perfect field, the Nisnevich localization of the \(\mathbb{A}^1\)-localization of the group completion of the presheaf taking \(U\in Sm/k\) to the space of framed correspondences from \(U\) to \(X\) is an equivalence of simplicial presheaves on \(Sm/k\). This theorem is the computational basis for the \(\infty\)-categorical elaboration of motivic infinite loop space machines (in the spirit of Segal's \(\Gamma\)-spaces) due to \textit{E. Elmanto} et al. [Camb. J. Math. 9, No. 2, 431--549 (2021; Zbl 07422194)].
The definition of framed correspondences is too technical for this review, and I also will not elaborate on the theory of `big framed motives' introduced by the authors. An important warning, though, is in order: to the contrary of the authors' claims, framed correspondences do \emph{not} form the morphism sets of a category. One usually composes correspondences via pullback, but the data of a framed correspondence (speicifically the closed subset \(Z\) of \(\mathbb A^n_X\) from Definition 2.1) is not preserved by this construction. See [loc. cit., Paragraph 1.3.7] for an elaboration of this point, noting that those authors refer to Garkusha-Panin's framed correspondences as \emph{equationally} framed correspondences. This categorical frustration does not impact the arguments in the reviewed paper.
Reviewer: Kyle Ormsby (Portland)Density of rational points on a family of del Pezzo surfaces of degree one (with an appendix by Jean-Louis Colliot-Thélène)https://zbmath.org/1491.140352022-09-13T20:28:31.338867Z"Desjardins, Julie"https://zbmath.org/authors/?q=ai:desjardins.julie"Winter, Rosa"https://zbmath.org/authors/?q=ai:winter.rosaThe paper under review concerns the density of rational points on certain del Pezzo surfaces of degree 1 over fields of characteristic zero, but really the motivation comes from the problem of unirationality over non-closed fields, say \(k\). Here, it is natural to restrict to \(k\)-minimal del Pezzo surfaces, and then unirationality is essentially open only for degree 1 surfaces. More precisely, the Picard number (over \(k\)) can only be 1 or 2 for these surfaces, and in the latter case unirationality follows from [\textit{J. Kollár} and \textit{M. Mella}, Am. J. Math. 139, No. 4, 915--936 (2017; Zbl 1388.14096)], but the Picard number 1 case seems completely open. It thus comes naturally to consider the problem of density of rational points of del Pezzo surfaces of degree 1 (and Picard number 1) as a test case, since this is clearly implied by unirationality. Previous (unconditional) results in this direction include [\textit{A. Várilly-Alvarado}, Algebra Number Theory 5, No. 5, 659--690 (2011; Zbl 1276.11114)] and [\textit{C. Salgado} and \textit{R. van Luijk}, Adv. Math. 261, 154--199 (2014; Zbl 1296.14018)], but there are also conditional results using root numbers by the first author in [J. Lond. Math. Soc., II. Ser. 99, No. 2, 295--331 (2019; Zbl 1459.11141)].
In detail, the authors consider the degree 6 surfaces \(S\) in weighted projective space \(\mathbb P[1,1,2,3]\) given by \[y^2 = x^3 + az^6+bz^3w^3+cw^3 \tag{1}\] where \(a,b,c\in k\) with \(ab\neq 0\) and \(4ac\neq b^2\) (the last condition is not stated explicitly in the paper, but together they ensure that the equations indeed define a del Pezzo surface of degree 1).
The main result states the sufficient condition that \(S(k)\) is Zariski dense if there are \(z,w\neq 0\) such that the elliptic curve resulting upon substituting into (1) has positive rank over \(k\). Moreover, this condition is also necessary if \(k\) is of finite type over \(\mathbb Q\).
To prove the density, the authors argue with the (isotrivial) elliptic surface defined by (1) (which has torsion-free Mordell-Weil group as a consequence of the above conditions). Formally, it is obtained from \(S\) by blowing up the base point \((0:0:1:1)\) of the linear system \(|-K_S|\). Starting from a non-trivial point \(R\) in a fibre of (1), the authors exhibit an auxilliary (possibly reducible) curve \(C_R\) which serves as trisection of the elliptic fibration. The curve is chosen as a special member of the linear system \(|-3K_S|\) which is singular at \(R\) and at two other points. It was pointed out by Kollàr that \(C_R\) can also be derived from a cubic surface which is \(3:1\) dominated by \(S\) in a natural way.
If the fibre \(F\) in question is smooth, then the authors prove that \(R\) can be chosen in such a way that
\begin{itemize}
\item \(C_R\) contains an irreducible component over \(k\) which gives a section of the fibration, or
\item \(C_R\) is geometrically integral of genus 0, or
\item \(C_R\) is geometrically integral of genus 1 for \(R\) varying over an open subset of \(F\).
\end{itemize}
The proof of the potential density in the main theorem then proceeds by a careful case-by-case analysis. The necessity of the given condition, over fields of finite type over \(\mathbb Q\), builds on a theorem Colliot-Thélène which improves upon Merel's theorem for bounding the torsion in families of elliptic curves.
The paper concludes with three interesting examples illustrating the techniques developed.
Reviewer: Matthias Schütt (Hannover)Diophantine stabilityhttps://zbmath.org/1491.140362022-09-13T20:28:31.338867Z"Mazur, Barry"https://zbmath.org/authors/?q=ai:mazur.barry"Rubin, Karl"https://zbmath.org/authors/?q=ai:rubin.karlLet \(K\) be a number field, \(\overline{ K}\) a separable closure of \(K\), \(V\) an irreducible variety over \(K\) and \(L\) a field containing \(K\). The authors say that \(V\) is {\em diophantine-stable for \(L/K\) } if \(V(L)=V(K)\). If \(\ell\) is a rational prime, they say that \(V\) is {\em \(\ell\)-diophantine stable} if for every positive integer \(n\), and every finite set \(\Sigma\) of places of \(K\), there are infinitely many cyclic extensions \(L/K\) of degree \(\ell^n\), completely split at all places \(v\in\Sigma\), such that \(V\) is diophantine-stable for \(L/K\). \par The first main result deals with a simple abelian variety \(A\) over \(K\). Assume that all \({\overline K}\)-endomorphisms of \(A\) are defined over \(K\). Then there is a set \(S\) of rational primes with positive density such that \(A\) is \(\ell\)-diophantine-stable over \(K\) for every \(\ell\in S\).
The second main result, which is a consequence of the first one, deals with an irreducible curve \(X\) over \(K\). Assume that the normalisation and completion \(\tilde X\) of \(X\) has genus \(\ge 1\) and that all \(\overline{K}\)-endomorphisms of the Jacobian of \(\tilde X\) are defined over \(K\); then there is a set \(S\) of rational primes with positive density such that \(X\) is \(\ell\)-diophantine-stable over \(K\) for every \(\ell\in S\).
From the second result the authors deduce the following two statements, the first one by applying repeatedly their result to the modular curve \(X_0(p)\), the second one to an elliptic curve over \({\mathbb Q}\) of positive rank. Let \(p\ge 23\) with \(p\not\in\{37,43,67,163\}\); then there are uncountably many pairwise non-isomorphic subfields \(L\) of \(\overline{\mathbb Q}\) such that no elliptic curve defined over \(L\) possesses an \(L\)-rational subgroup of ordre \(p\). And finally the authors prove that for every prime \(p\), there are uncountably many pairwise non-isomorphic totally real fields \(L\) of algebraic numbers in \({\mathbb Q}_p\) over which the following two statements both hold: (i) There is a diophantine definition of \(\mathbb Z\) in the ring of integers \({\mathcal O}_L\) of \(L\); in particular Hilbert's Tenth Problem has a negative answer for \({\mathcal O}_L\). (ii) There exists a first-order definition of the ring \(\mathbb Z\) in \(L\); the first-order theory for such fields \(L\) is undecidable.
The appendix by M. Larsen is devoted to the proof of the following result. Let \(A\) be a simple abelian variety defined over \(K\) such that \({\mathcal E}:={\mathrm{End}}_K(A)= {\mathrm{End}}_{\overline K}(A)\). Let \({\mathcal R}\) denote the center of \(\mathcal E\) and \({\mathcal M}={\mathcal R}\otimes{\mathbb Q}\). There is a positive density set \(S\) of rational primes such that for every prime \(\lambda\) of \({\mathcal M}\) lyning above \(S\) we have: (i) there is a \(\tau_0\in G_{K^{\mathrm ab}}\) such that \(A[\lambda]^{(\tau_0)}=0\), and (ii) there is a \(\tau_1\in G_{K^{\mathrm ab}}\) such that \(A[\lambda]/(\tau_1-1)A[\lambda]\) is a simple \({\mathcal E}/\lambda\)-module.
Reviewer: Michel Waldschmidt (Paris)On the maximality of hyperelliptic Howe curves of genus 3https://zbmath.org/1491.140372022-09-13T20:28:31.338867Z"Ohashi, Ryo"https://zbmath.org/authors/?q=ai:ohashi.ryo.1Summary: In this paper, we study a Howe curve \(C\) in positive characteristic \(p \geq 3\) which is of genus 3 and is hyperelliptic. We will show that if \(C\) is superspecial, then its standard form is maximal or minimal over \(\mathbb{F}_{p^2}\) without taking its \(\mathbb{F}_{p^2} \)-form.On the BMY inequality on surfaceshttps://zbmath.org/1491.140382022-09-13T20:28:31.338867Z"Terzi, Sadık"https://zbmath.org/authors/?q=ai:terzi.sadikIn recent years there has been some interest in understanding the geography of surfaces of general type over an algebraically closed field \(k\) of positive characteristic; see e.g. [\textit{J. Junmyeong}, Mich. Math. J. 59, No. 1, 169--178 (2010; Zbl 1195.14052); \textit{G. Urzúa}, Duke Math. J. 166, No. 5, 975--1004 (2017; Zbl 1390.14105); \textit{J. Kirti}, Eur. J. Math. 6, No. 4, 1111--1175 (2020; Zbl 07335100)]. In these works, there is a special focus on the validity of some sort of Bogomolov-Miyaoka-Yau (BMY) type of inequality, which is a strong constraint on Chern numbers in characteristic zero. We know that it may be wildly violated: Given any \(r\geq 2\) and \(k\) of characteristic \(p\), there are minimal surfaces of general type \(S\) with \(c_1^2(S)/c_2(S)\) is arbitrarily close to \(r\) and reduced Picard scheme. The main result of the present paper is to show a general BMY inequality which involves a nonnegative term whose boundedness implies a bound for \(c_1^2/c_2\). More precisely, let \(S\) be an ordinary surface that admits a generically ordinary and semi-stable fibration \(S \to C\) of genus \(g \geq 2\), where \(C\) is a nonsingular projective curve of genus \(q \geq 1\). Then \[c_1^2(S) \leq 3 c_2(S) + \frac{12}{p-1} h^1(B^1_{S|_C}) -4 \delta,\] where \(\delta\) is the total number of singular points in fibers of \(S \to C\), and \(B^1_{S|_C}\) is a sheaf on \(S^{(p)}\) that sits in the short exact sequence \(0 \to \mathcal{O}_{S^{(p)}} \to F_* \mathcal{O}_S \to B^1_{S|_C} \to 0\), where \(F \colon S \to S^{(p)}\) is the relative Frobenius morphism. If the fibration is ordinary, then \(h^1(B^1_{S|_C})\) vanishes.
Reviewer: Giancarlo Urzúa (Santiago de Chile)Derived non-Archimedean analytic Hilbert spacehttps://zbmath.org/1491.140392022-09-13T20:28:31.338867Z"António, Jorge"https://zbmath.org/authors/?q=ai:antonio.jorge"Porta, Mauro"https://zbmath.org/authors/?q=ai:porta.mauroSummary: In this short paper, we combine the representability theorem introduced in
[\textit{M. Porta} and \textit{T. Y. Yu}, ``Representability theorem in derived analytic geometry'', Preprint, \url{arXiv:1704.01683}; ``Derived Hom spaces in rigid analytic geometry'', Preprint, \url{arXiv:1801.07730}] with the theory of derived formal models introduced in
[\textit{J. António}, ``Derived \(\mathcal{O}_k\)-adic geometry and derived Raynaud localization theorem'', Preprint, \url{arXiv:1805.03302}]
to prove the existence representability of the derived Hilbert space \(\mathbf{R}\mathrm{Hilb}(X)\) for a separated \(k\)-analytic space \(X\). Such representability results rely on a localization theorem stating that if \(\mathfrak{X}\) is a quasi-compact and quasi-separated formal scheme, then the \(\infty \)-category \(\mathrm{Coh}^-(\mathfrak{X}^{\mathrm{rig}})\) of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the \(\infty \)-category \(\mathrm{Coh}^-(\mathfrak{X})\). Along the way, we prove several results concerning the \(\infty \)-categories of formal models for almost perfect modules on derived \(k\)-analytic spaces.Lifting low-dimensional local systemshttps://zbmath.org/1491.140402022-09-13T20:28:31.338867Z"De Clercq, Charles"https://zbmath.org/authors/?q=ai:de-clercq.charles"Florence, Mathieu"https://zbmath.org/authors/?q=ai:florence.mathieuSummary: Let \(k\) be a field of characteristic \(p>0\). Denote by \(\mathbf{W}_r(k)\) the ring of truntacted Witt vectors of length \(r \ge 2\), built out of \(k\). In this text, we consider the following question, depending on a given profinite group \(G\). \(\mathcal Q(G)\): Does every (continuous) representation \(G\longrightarrow \mathrm{GL}_d(k)\) lift to a representation \(G\longrightarrow \mathrm{GL}_d(\mathbf{W}_r(k))\)? We work in the class of cyclotomic pairs (Definition 4.3), first introduced in our paper [``Smooth profinite groups. I: geometrizing Kummer theory'', Preprint, \url{arXiv:2009.11130}] under the name ``smooth profinite groups''. Using Grothendieck-Hilbert' theorem 90, we show that the algebraic fundamental groups of the following schemes are cyclotomic: spectra of semilocal rings over \(\mathbb{Z}[\frac{1}{p}]\), smooth curves over algebraically closed fields, and affine schemes over \(\mathbb{F}_p\). In particular, absolute Galois groups of fields fit into this class. We then give a positive partial answer to \(\mathcal Q(G)\), for a cyclotomic profinite group \(G\): the answer is positive, when \(d=2\) and \(r=2\). When \(d=2\) and \(r=\infty \), we show that any 2-dimensional representation of \(G\) \text{stably} lifts to a representation over \(\mathbf{W}(k)\): see Theorem 6.1. When \(p=2\) and \(k=\mathbb{F}_2\), we prove the same results, up to dimension \(d=4\). We then give a concrete application to algebraic geometry: we prove that local systems of low dimension lift Zariski-locally (Corollary 6.3).Arithmetic intersections of modular geodesicshttps://zbmath.org/1491.140412022-09-13T20:28:31.338867Z"Darmon, Henri"https://zbmath.org/authors/?q=ai:darmon.henri"Vonk, Jan"https://zbmath.org/authors/?q=ai:vonk.janThe authors write: ``The goal of this note is to propose an arithmetic intersection theory for modular geodesics, attaching to a pair of such geodesics certain numerical invariants that are rich enough to (ostensibly) generate class fields of real quadratic fields. The predicted algebraicity of these quantities is a by-product of the approach of \textit{H. Darmon} and \textit{J. Vonk} [Duke Math. J. 170, No. 1, 23--93 (2021; Zbl 1486.11137)] to explicit class field theory based on the RM values of rigid meromorphic cocycles, but avoids the latter notion and offers a somewhat complementary perspective.''
Here ``RM'' stands for real multiplication and a ``modular geodesic'' is the image on \(\mathrm{SL}_{2}(\mathbb{Z})\setminus\mathfrak{H}\) of the oriented geodesic joining the two roots of an indefinite primitive integral binary quadratic form on the upper half-plane \(\mathfrak{H}\).
For any pair of distinct closed geodesics \((\gamma_{1}, \gamma_{2})\) on \(\Gamma\setminus\mathfrak{H}\), where \(\Gamma\) is an arithmetic subgroup of \(\mathrm{SL}_{2}(\mathbb{R})\) arising from the multiplicative group of an order in an indefinite quaternion algebra \(B\) over \(\mathbb{Q}\), the authors define an arithmetic intersection \(\gamma_{1} \star \gamma_{2}\). Given a rational prime \(p\) not dividing the discriminant of \(B\) and an order \(R\) of \(B\) of discriminant prime to \(p\), let \(\Gamma_{p}:=(R[1/p])_{1}^{\star}\) (here \(O_{1}^{\star}\) stands for the group of the elements of an order \(O\) of reduced norm one), then a \(p\)-arithmetic intersection \((\gamma_{1} \star \gamma_{2})_{\Gamma_{p}}\) is defined as a certain infinite \(p\)-adically absolutely convergent product.
At last conjecturally certain ``incoherent intersection numbers'' attached to a pair of RM divisors are defined and their conjectural relation to compatible systems of geodesics on an ``incoherent collection'' of Shimura curves is briefly discussed. The authors state seven conjectures and ask a few questions concerning the hypothetical algebraic properties of the defined intersections.
Reviewer: B. Z. Moroz (Bonn)Application of automorphic forms to lattice problemshttps://zbmath.org/1491.140422022-09-13T20:28:31.338867Z"Düzlü, Samed"https://zbmath.org/authors/?q=ai:duzlu.samed"Krämer, Juliane"https://zbmath.org/authors/?q=ai:kramer.julianeThe paper under review is motivated by the work of de Boer, Ducas, Pellet-Mary, and Wesolowski on self-reducibility of ideal-SVP via Arakelov random walks [\textit{D. Micciancio} (ed.) and \textit{T. Ristenpart} (ed.), Advances in cryptology -- CRYPTO 2020. 40th annual international cryptology conference, CRYPTO 2020. Proceedings. Part II. Cham: Springer. 243--273 (2020; Zbl 07332292)]. More preceisely, the authors show a worst-case to average-case reduction for ideal lattices and explain their approach how the steps are reproduced for module lattices of a fixed rank over some number field. Two major distinctions in their approach are that for higher rank module lattices, the notion of Arakelov divisors is replaced by adèles and Fourier analysis is substituted by the notion of automorphic forms.
Note that subject to the Riemann hypothesis, the worst-case to average-case convergence is analyzed in terms of the Fourier series. Thereafter, the worst-case shortest vector problem is as hard as the averagecase shortest vector problem.
Reviewer: Sami Omar (Sukhair)Investigation of automorphism group for code associated with optimal curve of genus threehttps://zbmath.org/1491.140432022-09-13T20:28:31.338867Z"Malygina, E. S."https://zbmath.org/authors/?q=ai:malygina.e-sSummary: The main result of this paper is contained in two theorems. In the first theorem, it is proved that the mapping \(\lambda: \mathcal{L}(mP_\infty) \rightarrow \mathcal{L}(mP_\infty)\) has the multiplicative property on the corresponding Riemann-Roch space associated with the divisor \(mP_\infty\) which defines some algebraic-geometric code if the number of points of degree one in the function field of genus three optimal curve over finite field with a discriminant \(\lbrace -19, -43, -67, -163 \rbrace\) has the lower bound \(12m/(m-3)\). Using an explicit calculation with the valuations of the pole divisors of the images of the basis functions \(x,y,z\) in the function field of the curve via the mapping \(\lambda \), we have proved that the automorphism group of the function field of our curve is a subgroup in the automorphism group of the corresponding algebraic-geometric code. In the second theorem, it is proved that if \(m \geq 4\) and \(n>12m/(m-3)\), then the automorphism group of the function field of our curve is isomorphic to the automorphism group of the algebraic-geometric code associated with divisors \(\sum\limits_{i=1}^nP_i\) and \(mP_\infty \), where \(P_i\) are points of the degree one.Abel maps for nodal curves via tropical geometryhttps://zbmath.org/1491.140442022-09-13T20:28:31.338867Z"Abreu, Alex"https://zbmath.org/authors/?q=ai:abreu.alex-c"Andria, Sally"https://zbmath.org/authors/?q=ai:andria.sally"Pacini, Marco"https://zbmath.org/authors/?q=ai:pacini.marcoLet \(C\) be a nodal algebraic curve and let \(\pi : \mathcal{C} \to B\) be a smoothing. Then there is a degree \(d\) Abel map
\[
\alpha^d : \mathcal{C}^d = \mathcal{C} \times_B \cdots \times_B \mathcal{C} \longrightarrow \overline{\mathcal{J}}
\]
to Esteves compactified Jacobian \(\overline{\mathcal{J}}\). Over the generic fiber of \(\pi\) this is just the usual Abel map of a smooth curve and hence well-defined. Over the special fiber however, \(\alpha^d\) will in general not be defined everywhere, i.e. \(\alpha^d\) is only a rational map.
The main theorem of this article employs tropical and toric geometry to describe blow-ups of \(\mathcal{C}^d\) that resolve the degree \(d\) Abel map. Moreover, the authors show that in degree 1 the Abel map is always a morphism and for biconnected \(C\) it is even injective.
Recall that the tropical Jacobian \(J(X)\) of a metric graph \(X\) is defined as the set of degree \(0\) divisors modulo linear equivalence. In previous work [\textit{A. Abreu} and \textit{M. Pacini}, Proc. Lond. Math. Soc. (3) 120, No. 3, 328--369 (2020; Zbl 1453.14082)] have endowed \(J(X)\) with a polyhedral structure using so-called \emph{pseudo divisors}. The tropical degree \(d\) Abel map
\[
\alpha_d^\mathrm{trop} : X^d \longrightarrow J(X)
\]
is not a map of polyhedral complexes. In the present article the authors give a unimodular triangulation of \(X^d\) that makes \(\alpha_d^\mathrm{trop}\) a map of polyhedral complexes. Via toric geometry this subdivision corresponds to blow-ups of \(\mathcal{C}^d\) and the authors show that these blow-ups resolve \(\alpha^d\). The key to this is a result from [\textit{A. Abreu} et al., Mich. Math. J. 64, No. 1, 77--108 (2015; Zbl 1331.14035)] that gives a criterion for resolving \(\alpha^d\) locally around a node in terms of certain numerical invariants of \(C\). In the bulk of the paper these numerical invariants are redefined on the tropical side and it is shown that the triangulation satisfies the criterion on the tropical side (and hence on the algebraic side as well).
The authors remark that there is also an induced subdivision of \(X^d\) by pulling back the polyhedral structure of \(J(X)\) along the Abel map. They describe an implementation of an algorithm to compute this.
Reviewer: Felix Röhrle (Frankfurt am Main)Curves with prescribed symmetry and associated representations of mapping class groupshttps://zbmath.org/1491.140452022-09-13T20:28:31.338867Z"Boggi, Marco"https://zbmath.org/authors/?q=ai:boggi.marco"Looijenga, Eduard"https://zbmath.org/authors/?q=ai:looijenga.eduard-j-nA classical result from Hurwitz asserts that the endomorphism ring of the Jacobian of a very general curve \(C\) of genus at least 2 is the smallest possible, i.e. it is \(\mathbb{Z}\). \textit{S. Lefschetz} [Am. J. Math. 50, 159--166 (1928; JFM 54.0410.02)] proved the same in the case of hyperelliptic curves. \textit{C. Ciliberto} et al. [J. Algebr. Geom. 1, No. 2, 215--229 (1992; Zbl 0806.14020)] studied curves whose Jacobian has endomorphism ring larger than \(\mathbb{Z}\). Finally, \textit{Y. G. Zarhin} [Math. Proc. Camb. Philos. Soc. 136, No. 2, 257--267 (2004; Zbl 1058.14064)] considered curves with automorphisms (of a specific type) and showed that the endomorphism ring of their Jacobians is as smallest as possible.
In the paper under review, the authors show that such a minimality property still holds for curves endowed with an action of a given (but arbitrary) finite group \(G\). Indeed, they show that \(\text{End}_{\mathbb{Q}}(JC)\cong \mathbb{Q}[G]\) for a very general \(G\)-curve, with quotient curve of genus at least 3.
As an application, they also obtain interesting consequences on the natural representation of the centralizer of \(G\) in \(\text{Mod}(C)\), \(\rho_G: \text{Mod}(C)^G\rightarrow \text{Sp}(H^1(C,\mathbb{Q}))^G\), where \(\text{Sp}(H^1(C,\mathbb{Q}))^G\) stands for the centralizer of \(G\) in \(\text{Sp}(H^1(C,\mathbb{Q}))\), regarded as a virtual
linear representation the mapping class group \(\text{Mod}(C/G)\). Indeed, let \(X(\mathbb{Q}[G])\) be the set of rational irreducible characters of \( G \), take the isomorphism \[
\text{Sp}(H^1(C,\mathbb{Q}))^G\cong \prod_{\chi \in X(\mathbb{Q}[G]) }\text{Sp}(H^1(C,\mathbb{Q})_{\chi})^G,
\]
which mirrors the isotypical decomposition of the Jacobian of a \(G\)-curve \(C\), and denote by \(Mon^0(C)\) the identity component of the Zariski closure of the
image of \( \rho_G \) in \(\text{Sp}(H^1(C,\mathbb{Q}))^G\) and by \(\text{Mon}^0(C)_\chi\) the projection of \(\text{Mon}^0(C)\) to the factor \( \text{Sp}(H^1(C,\mathbb{Q})_{\chi})^G \). Then, the authors deduce different interesting properties on \(\text{Mon}^0(C)\) and on its factors \(\text{Mon}^0(C)_\chi\).
Reviewer: Irene Spelta (Pavia)Divisorial motivic zeta functions for marked stable curveshttps://zbmath.org/1491.140462022-09-13T20:28:31.338867Z"Brandt, Madeline"https://zbmath.org/authors/?q=ai:brandt.madeline"Ulirsch, Martin"https://zbmath.org/authors/?q=ai:ulirsch.martinSummary: We define a divisorial motivic zeta function for stable curves with marked points, which agrees with Kapranov's motivic zeta function when the curve is smooth and unmarked. We show that this zeta function is rational and give a formula in terms of the dual graph of the curve.A Sylvester-Gallai theorem for cubic curveshttps://zbmath.org/1491.140472022-09-13T20:28:31.338867Z"Cohen, Alex"https://zbmath.org/authors/?q=ai:cohen.alex"de Zeeuw, Frank"https://zbmath.org/authors/?q=ai:de-zeeuw.frankThe theorem of Sylvester-Gallai, a classic result of combinatorial geometry, states that for any finite set \(A\) of points in \(\mathbb{R}^2\), not contained in a line, there exists a straight line that intersects \(A\) in exactly two points. It is conjectured that for any finite point set \(A \subset \mathbb{R}^2\) that is not contained in an algebraic curve of degree \(d\) there exist a -- not necessarily irreducible -- algebraic curve of degree \(d\) that intersects \(\mathbb{P}\) in precisely \(d(d+3)/2\) points (the number of points that typically determine a curve of degree \(d\)).
This conjecture was shown to be true for \(d = 2\) by \textit{J. A. Wiseman} and \textit{P. R. Wilson} in [Discrete Comput. Geom. 3, No. 4, 295--305 (1988; Zbl 0643.51009)]. Since then, several other proofs for the case \(d = 2\) have appeared. This article adds a new proof that allows for a generalization to cubic curves and sufficiently large point sets. While details of their proof are a bit involved and complicated, the basic idea is straightforward and elegant: In the projective space of cubic curves a plane is determined by requiring the cubics to pass through seven suitably chosen points. The dual version of the original Sylvester-Gallai theorem then guarantees existence of a point in that space that leads to a solution cubic.
The authors write quite openly about the caveats of their approach and also explain reasons for some of its weak points:
\begin{itemize}
\item The point set \(A\) is required to have a cardinality of at least \(250\). This number is certainly not optimal and could be reduced. However, by using only ideas from this article, it seems impossible to get entirely rid of a restriction on the cardinality of \(A\).
\item Unlike the line and conic versions of the Sylvester-Gallai theorem, the authors cannot guarantee that the nine points determine a \emph{unique} cubic.
\item Generalization to higher degrees seem difficult. The authors outline a possible approach based on Sylvester-Gallai theorems for points and hyperplanes in higher dimensions. However, nothing seems to be known in that regard and it is entirely possible that these theorems fail to be true.
\end{itemize}
Reviewer: Hans-Peter Schröcker (Innsbruck)The Severi problem for abelian surfaces in the primitive casehttps://zbmath.org/1491.140482022-09-13T20:28:31.338867Z"Zahariuc, Adrian"https://zbmath.org/authors/?q=ai:zahariuc.adrianLet \((A,\mathcal{L})\) be a general \((1,d)\)-polarized complex abelian surface and denote by \(\beta \in \operatorname{NS}(A)\) the class of \(\mathcal{L}\) in the Neron-Severi group.
Let \(V_g(A,\beta)\), for \(3 \leq g \leq d + 1\), be the Severi variety that parametrizes those unramified maps \(f \colon C \rightarrow A\text{,}\) from a smooth connected genus \(g\) curve \(C\), which have the property that \(f_*[C] = \beta\text{.}\)
In the article under review, the author's main result identifies the irreducible components of \(V_g(A,\beta)\). This answers Question 4.2 from \textit{A. L. Knutsen} et al. [J. Reine Angew. Math. 749, 161--200 (2019; Zbl 1439.14021)].
In proving this result, the author builds on the techniques of \textit{J. Bryan} and \textit{N. C. Leung} [Duke Math. J. 99, No. 2, 311--328 (1999; Zbl 0976.14033)], \textit{X. Chen} [Math. Ann. 324, No. 1, 71--104 (2002; Zbl 1039.14019)] and \textit{A. Zahariuc} [Proc. Lond. Math. Soc. (3) 119, No. 6, 1431--1463 (2019; Zbl 07174775)].
A key point is to specialize the given abelian surface to a product of elliptic curves. In particular, the author develops a criterion that characterizes those primitive class staple maps to a product of elliptic curves that deform to nearby surfaces. This is the author's main technical result (Theorem 4.3).
Reviewer: Nathan Grieve (Kingston)Symmetric Galois groups under specializationhttps://zbmath.org/1491.140492022-09-13T20:28:31.338867Z"Monderer, Tali"https://zbmath.org/authors/?q=ai:monderer.tali"Neftin, Danny"https://zbmath.org/authors/?q=ai:neftin.dannyConsider a polynomial \(f(t,x) \in \mathbb{Q}(t)[x]\) with coefficients depending on \(t\), and let \(G\) be the Galois group of the polynomial. For all but finitely many specializations \(t \mapsto t_0 \in \mathbb{Q}\), the Galois group \(\text{Gal}(f(t_0,x), \mathbb{Q})\) is a subgroup of \(G\); and for infinitely many \(t_0\), that group is just \(G\). The question in this article is what proper subgroups of \(G\) are \(\text{Gal}(f(t_0,x), \mathbb{Q})\) for infinitely many \(t_0 \in \mathbb{Q}\). The answer to that question is given in the main Theorem 1.1, according to which, if \(G\) is \(A_n\) or \(S_n\) for \(n\) greater than a certain constant \(N_1\), and \(\text{Gal}(f(t_0,x), \mathbb{Q}) \approx H\) for infinitely many \(t_0 \in \mathbb{Q}\), then either i) \(H = A_n\) or \(S_n\), or ii) \(H = A_{n-1}\) or \(S_{n-1}\), or iii) \(A_{n-2} \lneq H \lneq S_{n-2} \times S_2\). The way to prove Theorem 1.1 is to classify low genus covers with monodromy \(A_n\) or \(S_n\). This is made in Theorem 1.2 by an analysis of the transitivity of the action of \(H\) on unordered sets, and Theorem 1.1 follows using Faltings' theorem.
Reviewer: José Javier Etayo (Madrid)Curves with rational families of quasi-toric relationshttps://zbmath.org/1491.140502022-09-13T20:28:31.338867Z"Kloosterman, Remke"https://zbmath.org/authors/?q=ai:kloosterman.remke-nanneSummary: We investigate which plane curves admit rational families of quasi-toric relations. This extends previous results of \textit{A. Takahashi} and \textit{H. Tokunaga} [``Representations of divisors on hyperelliptic curves, Gröbner bases and plane curves with quasi-toric relations'', Preprint, \url{arXiv:2102.05794}] in the positive case and of the author [Math. Ann. 367, No. 1--2, 755--783 (2017; Zbl 1370.14031)] in the negative case.Betti numbers of Brill-Noether varieties on a general curvehttps://zbmath.org/1491.140512022-09-13T20:28:31.338867Z"Felisetti, Camilla"https://zbmath.org/authors/?q=ai:felisetti.camilla"Fontanari, Claudio"https://zbmath.org/authors/?q=ai:fontanari.claudioSummary: We compute the rational cohomology groups of the smooth Brill-Noether varieties \(G^r_d(C)\), parametrizing linear series of degree \(d\) and dimension exactly \(r\) on a general curve \(C\). As an application, we determine the whole intersection cohomology of the singular Brill-Noether loci \(W^r_d(C)\), parametrizing complete linear series on \(C\) of degree \(d\) and dimension at least \(r\).Elliptic curve involving subfamilies of rank at least 5 over \(\mathbb{Q}(t)\) or \(\mathbb{Q}(t,k)\)https://zbmath.org/1491.140522022-09-13T20:28:31.338867Z"Youmbai, Ahmed El Amine"https://zbmath.org/authors/?q=ai:youmbai.ahmed-el-amine"Uludağ, Muhammed"https://zbmath.org/authors/?q=ai:uludag.muhammed-a"Behloul, Djilali"https://zbmath.org/authors/?q=ai:behloul.djilaliThe authors provide one- and two-parameter families of elliptic curves with large Mordell-Weil rank. In fact, they exhibit elliptic curves over \(\mathbb Q(t)\) with rank at least 5 that are either induced by the edges of a rational cuboid or by Diophantine triples. Using careful specialization of \(t\), examples of elliptic curves over \(\mathbb Q\) with rank 8, 9, 10 and 11 are given.
Reviewer: Mohammad Sadek (New Cairo)The hyperelliptic theta map and osculating projectionshttps://zbmath.org/1491.140532022-09-13T20:28:31.338867Z"Bolognesi, Michele"https://zbmath.org/authors/?q=ai:bolognesi.michele"Vargas, N. F."https://zbmath.org/authors/?q=ai:vargas.n-fLet \(C\) be a hyperelliptic curve of genus \(g\geq 3\) and \(\mathcal{SU}_C(r)\) the (coarse) moduli space of semistable vector bundles of rank \(r\) with trivial determinant on \(C,\) and \(D\) an effective divisor of degree \(g\) on \(C\). Consider the theta map \[ \theta : \mathcal{SU}_C(r) \dashrightarrow |r \Theta | . \]
In the present paper the authors give a geometric description of the theta map for \(r=2.\) More precisely, they prove that there exists a fibration \[p_D:\mathcal{SU}_C(2) \dashrightarrow |2 \Theta |\cong \mathbb{P}^g\] whose general fiber is birational to \(\mathcal{M}_{0,2g}^{GIT} \) and the theta map restricted to each of these fibers is a \(2\)-to-\(1\) osculating projection up to composition with a birational map. Moreover, they prove that the ramification locus of the theta map has an irreducible component birational to a fibration in Kummer varieties of dimention \(g-1\) over \(|2D|\cong \mathbb{P}^g.\)
Reviewer: Rick Rischter (Itajubá)Singularities of normal quartic surfaces. I: (\(\mathrm{char}= 2\))https://zbmath.org/1491.140542022-09-13T20:28:31.338867Z"Catanese, Fabrizio"https://zbmath.org/authors/?q=ai:catanese.fabrizioIn this very interesting paper, the author shows, using rather elementary methods, that the maximal number of singular points of a normal quartic surface in \(\mathbb{P}^{3}_{\mathbb{K}}\), where \(\mathbb{K}\) is an algebraically closed field of characteristic 2, is at most 16, and the author provides easy examples which lead to the conjecture that in fact the maximal number is at most 14. An explicit example of a quartic surface with 14 singular points has the following presentation:
\(X:=\{(z,x_{1},x_{2},x_{3}) \, : z^{2}(x_{1}x_{2}+x_{3}^{2})+(y_{3}+x_{1})(y_{3}+x_{2})y_{3}(y_{3}+x_{1}+x_{2}) = 0, \,\, y_{3}=a_{3}x_{3}+a_{2}x_{2}+a_{1}x_{1} \text{ with } a_{3}\neq 0, a_{1},a_{2},a_{3} \text{ general}\}\).
More precisely, the author shows, among others, that if \(X\) is a normal quartic surface defined over an algebraically closed field \(\mathbb{K}\) of characteristic 2, then:
a) If \(X\) has a point of multiplicity 3, then \(|\mathrm{Sing}(X)|\leq 7\),
b) If \(X\) has a double point \(P\) such that the projection with centre \(P\) is an inseparable double cover of \(\mathbb{P}^{2}_{\mathbb{K}}\), then \(\mathrm{Sing}(X) \leq 16\), and there exists a case with \(|\mathrm{Sing}(X)|=14\).
In the second part of the work, written together with \textit{M. Schuett} [``Singularities of normal quartic surfaces. II: \((\mathrm{char}=2)\)'', Preprint, \url{arXiv:2110.03078}], the authors show that in fact \(|\mathrm{Sing}(X)|\leq 14\) with equality only if the singular points are nodes and the minimal resolution of \(X\) is a supersingular \(K3\) surface.
Reviewer: Piotr Pokora (Kraków)Local dynamics of non-invertible maps near normal surface singularitieshttps://zbmath.org/1491.140552022-09-13T20:28:31.338867Z"Gignac, William"https://zbmath.org/authors/?q=ai:gignac.william"Ruggiero, Matteo"https://zbmath.org/authors/?q=ai:ruggiero.matteoIn the study of the dynamics of a dominant non-invertible holomorphic map germ \(f:(\mathbb{C}^2,0) \to (\mathbb{C}^2,0)\), a successful approach consists in investigating the dynamics of \(f\) on modifications of \((\mathbb{C}^2,0)\). Here a modification \(\pi:X_\pi \to (\mathbb{C}^2,0)\) is a proper holomorphic map that is an isomorphism over \(\mathbb{C}^2 \setminus \{0\}\), and one studies then the dynamics of the induced (meromorphic) map \(f_\pi:X_\pi \dashrightarrow X_\pi\) on the exceptional set \(\pi^{-1}\{0\}\). In this memoir the authors generalize many results to the singular case, namely replacing \((\mathbb{C}^2,0)\) by the germ of a normal surface singularity \((X,x_0)\).
A first main result is that the problematic situation of indeterminacy points for infinitely many powers \(f_\pi^n\) cannot occur (except in the very special case of a finite germ at a cusp singularity). Namely, for any modification \(\pi:X_\pi \to (X,x_0)\) one can find a modification \(\pi':X_{\pi'} \to (X,x_0) \) dominating \(\pi\) such that, if \(E\) is an exceptional divisor of \(\pi'\), then \(f_{\pi'}^n (E)\) is an indeterminacy point of the lift \(f_{\pi'}: X_{\pi'} \dashrightarrow X_{\pi'}\) for at most finitely many \(n\). Moreover \(X_{\pi'}\) can be chosen to have at most cyclic quotient singularities.
As in the smooth case, the strategy to prove this is analyzing the dynamics on a space that encodes all such modifications simultaneously, namely a suitable space \(\mathcal V_X\) of centered, rank one semivaluations on the local ring \(\mathcal O_{X,x_0}\), with the induced \(f_*:\mathcal V_X \to \mathcal V_X\). The result is roughly as follows: there is a subset \(S\subset \mathcal V_X\), homeomorphic to either a point, a closed interval or a circle, such that \(f_*(S) =S\) and for any quasimonomial valuation \(v\in \mathcal V_X\) we have that \(f_*^n(v) \to S\) as \(n \to \infty\). Its proof is the core of the paper, with main technical tool the construction of a suitable distance on \(\mathcal V_X\) and the study of its non-expanding properties.
Further, the authors derive three applications. The first is an `asymptotic functoriality' result, partially controlling the fact that the pull-back on the group of exceptional divisors of a modification \(\pi\) is in general not functorial. The second treats the sequence of attraction rates of a quasimonomial \(v\in \mathcal V_X\): it eventually satisfies an integral linear recursion relation (with a similar exception as before). The third says that the first dynamical degree of \(f\) is a quadratic integer.
It is also worth mentioning that, since the used techniques are valuative (rather than complex analytic), all results are in fact valid over an arbitrary field of characteristic zero, and that some results are even valid in positive characteristic.
Reviewer: Wim Veys (Leuven)Alpha invariant along curves for general polarizations of del Pezzo surfaces of degree 2https://zbmath.org/1491.140562022-09-13T20:28:31.338867Z"Won, Joonyeong"https://zbmath.org/authors/?q=ai:won.joonyeongSummary: For an arbitrary ample divisor \(A\) in smooth del Pezzo surface \(S\) of degree 2, we completely compute alpha invariant along curves when the ample divisor \(A\) is birational type.On the geometry of singular \(K3\) surfaces with discriminant 3, 4 and 7https://zbmath.org/1491.140572022-09-13T20:28:31.338867Z"Takatsu, Taiki"https://zbmath.org/authors/?q=ai:takatsu.taikiThe present article studies the three singular \(K3\) surfaces \(X_3\), \(X_4\) and \(X_7\), of discriminant \(3\), \(4\) and \(7\) respectively. These \(K3\) surfaces were explicitly constructed by \textit{T. Shioda} and \textit{H. Inose} [in: Complex Anal. algebr. Geom., Collect. Pap. dedic. K. Kodaira, 119--136 (1977; Zbl 0374.14006)].
The author shows that the \(K3\) surfaces \(X_3\), \(X_4\) and \(X_7\) admit a realization as a double cover of \(\mathbb{P}^2\) branched over a sextic curve with prescribed singularities. As a consequence, the three surfaces can be viewed as points in a \(3\)-dimensional moduli space of \(K3\) surfaces, namely the moduli space of \(U\oplus A_5^3\)-polarized \(K3\) surfaces.
Reviewer: Giacomo Mezzedimi (Hannover)Conics in sextic \(K3\)-surfaces in \(\mathbb{P}^4\)https://zbmath.org/1491.140582022-09-13T20:28:31.338867Z"Degtyarev, Alex"https://zbmath.org/authors/?q=ai:degtyarev.alexCounting the number of rational curves on complex \(K3\) surfaces is a classical and very hard problem. Since there are infinitely many rational curves on any \(K3\) surface \(X\), one can try to bound the number of rational curves up to a certain degree, relative to a fixed polarization for \(X\).
Some instances of this problem have a long history. \textit{B. Segre} [Q. J. Math., Oxf. Ser. 14, 86--96 (1943; Zbl 0063.06860)] showed that there are at most \(64\) lines on a smooth quartic surface (although his proof had a gap, which was fixed by \textit{S. Rams} and \textit{M. Schütt} [Math. Ann. 362, No. 1--2, 679--698 (2015; Zbl 1319.14042)]). Concerning polarizations of higher degree, \textit{A. Degtyarev} [Discrete Comput. Geom. 62, No. 3, 601--648 (2019; Zbl 1427.14077)] found sharp bounds for the number of lines on a \(2d\)-polarized \(K3\) surface \(X\subseteq \mathbb{P}^{d+1}\). Nevertheless, very little is known about the number of rational curves of higher degree on \(2d\)-polarized \(K3\) surfaces. In the case of conics, it is known that a quartic surface contains at most \(5016\) conics, but the best examples at the moment contain at most \(352\) and \(432\) conics (see \textit{Th. Bauer} [J. Reine Angew. Math. 464, 207--217 (1995; Zbl 0826.14020)]).
The main result of the present article proves a sharp bound for the number of conics on a sextic \(K3\) surface in \(\mathbb{P}^4\). More precisely, it is shown that a smooth sextic \(K3\) surface in \(\mathbb{P}^4\) contains at most \(285\) conics, and there are only three sextics with more than \(260\) conics.
The proof consists of a thorough analysis of possible configurations of conics on sextic \(K3\) surfaces, and it uses lattice-theoretical computations to decide whether a given configuration of conics really appears on some sextic \(K3\) surface.
Reviewer: Giacomo Mezzedimi (Hannover)On intersection cohomology and Lagrangian fibrations of irreducible symplectic varietieshttps://zbmath.org/1491.140592022-09-13T20:28:31.338867Z"Felisetti, Camilla"https://zbmath.org/authors/?q=ai:felisetti.camilla"Shen, Junliang"https://zbmath.org/authors/?q=ai:shen.junliang"Yin, Qizheng"https://zbmath.org/authors/?q=ai:yin.qizhengIrreducible holomorphic symplectic manifolds, that is simply connected compact Kähler manifolds \(M\) such that there exists a symplectic form \(\sigma \in H^0(M, \Omega_M^2)\) that generates all the other holomorphic forms, have been studied intensively since their introduction as part of the Beauville-Bogomolov decomposition. Since the generalisation of the decomposition theorem to klt spaces, cf. [\textit{A. Höring} and \textit{T. Peternell}, Invent. Math. 216, No. 2, 395--419 (2019; Zbl 07061101)], singular irreducible symplectic varieties appear as a natural extension of this important class of manifolds. In this paper the authors consider irreducible symplectic projective varieties that admit a Lagrangian fibration \(\pi: M \rightarrow B\), i.e. a fibration onto a normal projective variety \(B\) such that the general fibre is an abelian variety of dimension \(\frac{1}{2}\dim M\). In this case the perverse \(t\)-structure on the constructible derived category of \(B\) induces a filtration on the intersection cohomology \(\mbox{IH}^*(M, \mathbb C)\), and one denotes by \(^{\mathfrak p}\mbox{Ih}^{i,j}(\pi)\) the perverse numbers determined by the graded pieces of the filtration. The first main result of this paper is that the perverse numbers \(^{\mathfrak p}\mbox{Ih}^{i,j}(\pi)\) are a deformation invariant of Lagrangian fibrations if the second Betti number of the variety \(M\) is at least five. Moreover the authors show that \[ ^{\mathfrak p}\mbox{Ih}^{0,d}(\pi) = \ ^{\mathfrak p}\mbox{Ih}^{d,0}(\pi) \] vanishes for \(d\) odd and is equal to one for \(d\) even. As a consequence one obtains that the intersection cohomology \(\mbox{IH}^*(B, \mathbb C)\) on the base \(B\) is isomorphic to the cohomology of the projective space \(H^*(\mathbb P^{\frac{1}{2}\dim M}, \mathbb C)\). Note that while for smooth symplectic manifolds one expects that the base \(B\) of the Lagrangian fibration is actually a projective space, cp. [\textit{J.-M. Hwang}, Invent. Math. 174, No. 3, 625--644 (2008; Zbl 1161.14029)], there are examples of singular irreducible symplectic varieties where this is not the case [\textit{D. Matsushita}, Sci. China, Math. 58, No. 3, 531--542 (2015; Zbl 1317.14023)]. The authors also show that the image of the restriction map \[ \mbox{IH}^*(M, \mathbb C) \rightarrow H^*(M_b, \mathbb C) \] to a smooth fibre \(M_b\) of the Lagrangian fibration is generated by a relatively ample divisor class. \newline The second part of the paper concerns the relation between the perverse numbers and the Hodge numbers \(\mbox{Ih}^{i,j}(M)\) defined by the pure Hodge structure on \(\mbox{IH}^*(M, \mathbb C)\). If the variety \(M\) admits a resolution of singularities \(M' \rightarrow M\) such that the symplectic form \(\sigma\) pulls-back to a {\em symplectic} form on \(M'\), the authors show that the numbers coincide, i.e. one has \[ \mbox{Ih}^{i,j}(M) = \ ^{\mathfrak p}\mbox{Ih}^{i,j}(\pi) \] for all \(i,j\). As part of the proof they show that for symplectic varieties \(M\) admitting such a symplectic resolution the LLV algebra [\textit{E. Looijenga} and \textit{V. A. Lunts}, Invent. Math. 129, No. 2, 361--412 (1997; Zbl 0890.53030)] associated with the intersection cohomology is isomorphic to \(\mathfrak{so}(b_2(M)+2)\).
Reviewer: Andreas Höring (Nice)Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groupshttps://zbmath.org/1491.140602022-09-13T20:28:31.338867Z"Ito, Atsushi"https://zbmath.org/authors/?q=ai:ito.atsushi-mSummary: It is known that the automorphism group of a K-polystable Fano manifold is reductive. \textit{G. Codogni} and \textit{R. Dervan} [Ann. Inst. Fourier 66, No. 5, 1895--1921 (2016; Zbl 1370.32010)] constructed a canonical filtration of the section ring, called Loewy filtration, and conjectured that the filtration destabilizes any Fano variety with non-reductive automorphism group. In this note, we give a counterexample to their conjecture.K-stability of cubic fourfoldshttps://zbmath.org/1491.140612022-09-13T20:28:31.338867Z"Liu, Yuchen"https://zbmath.org/authors/?q=ai:liu.yuchen.1This paper studies the K-stability of cubic \(4\)-folds. K-stability is an algebraic stability condition which is used to construct moduli spaces of Fano varieties, which is known as the K-moduli theory.
It is expected that smooth Fano hypersurfaces are K-stable and admits Kähler-Einstein metrics. The main result of this paper shows that if \(X\) is a cubic hypersurface in \(\mathbb{P}^5\) (i.e. a cubic \(4\)-fold), then \(X\) is K-(semi/poly)stable iff \(X\) is GIT (semi/poly)stable. In particular, the K-moduli space parametrizing K-polystable \(\mathbb{Q}\)-Fano varieties admitting \(\mathbb{Q}\)-Gorenstein smoothings to smooth cubic \(4\)-fold is isomorphic to the GIT moduli space of cubic fourfolds. As corollaries, any smooth cubic \(4\)-fold is K-stable.
In fact, \textit{Y. Liu} and \textit{C. Xu} established a framework in [Duke Math. J. 168, No. 11, 2029--2073 (2019; Zbl 1436.14085)] which reduces the K-stability problem of cubic hypersurface to the ODP Gap conjecture of local volumes, namely, if \((x\in X)\) is a non-smooth klt singularity of dimension \(n\), then \(\widehat{\text{vol}}(x, X)\leq 2(n-1)^n\), and the equality holds iff \(x\) is an ordinary double point. In [loc. cit.], it was shown that the ODP Gap conjecture holds in dimension \(3\). Such conjecture is used to show that the K-semistable limits of cubic hypersurfaces are still hypersurfaces.
In this paper, instead of showing the ODP Gap conjecture in the full generality, the author shows the ODP Gap conjecture holds for local complete intersections (in arbitrary dimension), which turns out to be sufficient for the K-stability problem of cubic \(4\)-folds.
Reviewer: Chen Jiang (Shanghai)Delta invariants of projective bundles and projective cones of Fano typehttps://zbmath.org/1491.140622022-09-13T20:28:31.338867Z"Zhang, Kewei"https://zbmath.org/authors/?q=ai:zhang.kewei"Zhou, Chuyu"https://zbmath.org/authors/?q=ai:zhou.chuyuThe \(\delta\)-invariant (also known as the stability threshold) of a Fano variety characterizes its K-stability. It was proved by \textit{K. Fujita} and \textit{Y. Odaka} [Tohoku Math. J. (2) 70, No. 4, 511--521 (2018; Zbl 1422.14047)] and \textit{H. Blum} and \textit{M. Jonsson} [Adv. Math. 365, Article ID 107062, 57 p. (2020; Zbl 1441.14137)] that a Fano variety \(X\) is K-semistable (resp. uniformly K-stable) if and only if \(\delta(X) > 1\) (resp. \(\delta(X) \geq 1\)). If \(X\) is not uniformly K-stable, the \(\delta\)-invariant equals the greatest Ricci lower bound as shown in [\textit{I. A. Cheltsov} et al., Sel. Math., New Ser. 25, No. 2, Paper No. 34, 36 p. (2019; Zbl 1418.32015); \textit{R. J. Berman} et al., J. Am. Math. Soc. 34, No. 3, 605--652 (2021; Zbl 1487.32141)]. In this article, the authors compute \(\delta\)-invariant of \(\mathbb{P}^1\)-bundles and projective cones over a Fano variety \(V\).
Suppose \(V\) is a Fano variety of dimension \(n\) and Fano index \(\geq 2\). Let \(L = -\frac{1}{r}K_V\) be an ample line bundle for some rational number \(r>1\). Let \(\tilde{Y}:=\mathbb{P}_V(L^{-1} \oplus\mathcal{O}_V)\) be the \(\mathbb{P}^1\)-bundle as the compactification of the total space of \(L^{-1}\). Let
\[
\beta_0:= \left(\frac{n+1}{n+2}\cdot \frac{(r+1)^{n+2} - (r-1)^{n+2}}{(r+1)^{n+1} - (r-1)^{n+1}} - (r-1)\right)^{-1}.
\]
Theorem 1.1 states that
\[
\delta(\tilde{Y}) = \min \left\{\frac{\delta(V)r\beta_0}{1+\beta_0(r-1)}, \beta_0\right\}.
\]
Let \(Y\) be the projective cone over \(V\) with polarization \(L\). Then \(\tilde{Y}\to Y\) is a birational morphism that contracts the zero section \(V_0\) to the cone point in \(Y\). Theorem 1.4 implies that
\[
\delta(Y) = \frac{(n+2)r}{(n+1)(r+1)}\min\left\{1, \delta(V), \frac{n+1}{r}\right\}.
\]
In particular, if \(V\) is K-semistable then \(r\leq n+1\) by \textit{K. Fujita} [Am. J. Math. 140, No. 2, 391--414 (2018; Zbl 1400.14105)], so we have \(\delta(Y) = \frac{(n+2)r}{(n+1)(r+1)}\). Similar computations of \(\delta\)-invariants are also done for log Fano pairs \((\tilde{Y}, aV_0+bV_\infty)\) and \((Y, cV_\infty)\). Applications are included for computations of \(\delta\)-invariants of certain singular hypersurfaces, and the existence of conical Kähler-Einstein metrics on projective spaces with certain cone angle along a smooth Fano hypersurface.
Reviewer: Yuchen Liu (Evanston)On the structure of generalized polarized manifolds with relatively small second classhttps://zbmath.org/1491.140632022-09-13T20:28:31.338867Z"Lanteri, Antonio"https://zbmath.org/authors/?q=ai:lanteri.antonio"Tironi, Andrea Luigi"https://zbmath.org/authors/?q=ai:tironi.andrea-luigiA pair \((X, H)\), where \(X\) is a smooth projective variety of dimension \(n\) defined over the field of complex numbers and \(H\) is an ample line bundle on \(X\), is called a polarized manifold. Polarized manifolds have been extensively studied in the field using their invariants. Well-known invariants of the pairs \((X,H)\) are its degree \(d:=H^n\), its sectional genus \(g(H):=1+\frac{1}{2}(K_X+(n-1)H)H^{n-1}\), and its delta genus \(\Delta(H):=n+H^n-h^0(H)\). In recent years, many possible extensions to a vector bundle-like setting were explored by the first author and others defining different invariants.
In the first part of the paper under review, the authors investigate polarized surfaces \((S,H)\) using their degree \(d\) and their class \(m\) (that is the number of tangent hyperplanes to S contained in a general pencil of hyperplanes). Namely, the structure of surfaces \((S, H)\) with \(m -3d \leq 3\) is described. Particular attention is given to the case \(m-2d\leq 1\) (this assumption forces the surface to be ruled) providing an explicit description of the pair \((S,H)\). The approach is inspired by the first named author's previous works [Math. Nachr. 175, 199--207 (1995; Zbl 0874.14023)], where surfaces satisfying \(d<m<2d\) are classified, and [Arch. Math. 64, No. 4, 359--368 (1995; Zbl 0819.14014)], where the inequality \(m-3d\geq 6\) for properly elliptic surfaces is proven and surfaces with \(m-3d\leq 6\) are classified.
In the second part of the manuscript, the higher dimensional case is treated. Four-tuples \((X, \mathcal{E}, H, S)\), (where \(X\) is a \(n-\)dimensional manifold, \(\mathcal{E}\) is a vector bundle of rank between \(2\) and \(n-2\), \(H\) an ample line bundle on \(X\), and the assumption that the vector bundle \(\mathcal{E}\oplus H^{\oplus (n-r-2)}\) admits a section vanishing on the smooth surface \(S\) is satisfied), are classified for small values of the difference \(m_2 -d\) (here \(d=H_S^2\) and \(m_2\) is an invariant introduced by the two authors in a previous work).
Reviewer: Davide Fusi (Bluffton)The Kapustin-Witten equations and nonabelian Hodge theoryhttps://zbmath.org/1491.140642022-09-13T20:28:31.338867Z"Liu, Chih-Chung"https://zbmath.org/authors/?q=ai:liu.chih-chung"Rayan, Steven"https://zbmath.org/authors/?q=ai:rayan.steven"Tanaka, Yuuji"https://zbmath.org/authors/?q=ai:tanaka.yuujiSummary: Arising from a topological twist of \({\mathscr{N}}=4\) super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrised by \(t\in{\mathbb{P}}^1\). The parameter corresponds to a linear combination of two super charges in the twist. When \(t=0\) and the four-manifold is a compact Kähler surface, the equations become the Simpson equations, which was originally studied by Hitchin on a compact Riemann surface, as demonstrated independently in works of Nakajima and the third-named author. At the same time, there is a notion of \(\lambda \)-connection in the nonabelian Hodge theory of Donaldson-Corlette-Hitchin-Simpson in which \(\lambda\) is also valued in \({\mathbb{P}}^1\). Varying \(\lambda\) interpolates between the moduli space of semistable Higgs sheaves with vanishing Chern classes on a smooth projective variety (at \(\lambda =0)\) and the moduli space of semisimple local systems on the same variety (at \(\lambda =1)\) in the twistor space. In this article, we utilise the correspondence furnished by nonabelian Hodge theory to describe a relation between the moduli spaces of solutions to the equations by Kapustin and Witten at \(t=0\) and \(t \in{{\mathbb{R}}} \,{\setminus}\, \{ 0 \}\) on a smooth, compact Kähler surface. We then provide supporting evidence for a more general form of this relation on a smooth, closed four-manifold by computing its expected dimension of the moduli space for each of \(t=0\) and \(t \in{{\mathbb{R}}} \,{\setminus}\, \{ 0 \} \).Computing discrete invariants of varieties in positive characteristic. I: Ekedahl-Oort types of curveshttps://zbmath.org/1491.140652022-09-13T20:28:31.338867Z"Moonen, Ben"https://zbmath.org/authors/?q=ai:moonen.benSummary: We develop a method to compute the Ekedahl-Oort type of a curve \(C\) over a field \(k\) of characteristic \(p\) (which is the isomorphism type of the \(p\)-kernel group scheme \(J[p]\), where \(J\) is the Jacobian of \(C)\). Part of our method is general, in that we introduce the new notion of a Hasse-Witt triple, which re-encodes in a useful way the information contained in the Dieudonné module of \(J[p]\). For complete intersection curves we then give a simple method to compute this Hasse-Witt triple. An implementation of this method is available in Magma.Classification of abelian Nash manifoldshttps://zbmath.org/1491.140662022-09-13T20:28:31.338867Z"Bao, Yixin"https://zbmath.org/authors/?q=ai:bao.yixin"Chen, Yangyang"https://zbmath.org/authors/?q=ai:chen.yangyang"Zhao, Yi"https://zbmath.org/authors/?q=ai:zhao.yi.1Roughly speaking, a Nash manifold is an abstract space that is obtained by gluing finitely many affine Nash manifolds. An affine Nash manifold is a \(\mathscr{C}^\infty\)-submanifold of a euclidean space whose underlying set is a semi-algebraic set. The collection of Nash manifolds (resp. affine Nash manifolds) and the Nash mappings between them form a category. The group objects in this category are called the Nash groups (resp. affine Nash groups). A major motivation for studying affine Nash groups comes from the fact that any finite cover of a real algebraic group is always an affine Nash group but not necessarily a real algebraic group. Likewise, every orbit of a real algebraic group is naturally an affine Nash manifold but not necessarily a real algebraic variety.
A key concept here is the notion of an \textit{almost linear Nash group}, that is, a Nash group admitting a Nash representation with a finite kernel. A thorough study of almost linear Nash groups can be found in [\textit{B. Sun}, Chin. Ann. Math., Ser. B 36, No. 3, 355--400 (2015; Zbl 1322.22009)]. An affine Nash group is said to be \textit{complete} if it has no nontrivial connected almost linear Nash subgroup. An \textit{abelian Nash manifold} is a connected complete affine Nash group. The abelian Nash manifolds can be viewed as the Nash analogs of real abelian varieties.
In the present article, the authors classify completely the isomorphism classes of real abelian varieties as well as the isomorphism classes of abelian Nash manifolds. To state their results in a precise manner, we will introduce the notion of a ``real polarizable lattice.'' A pair \((V_0,\Lambda)\), where \(V_0\) is a finite dimensional real vector space, and \(\Lambda\) is a full lattice in \(V_0\otimes_{\mathbb{R}} \mathbb{C}\) such that
\[
\Lambda \subseteq \frac{1}{2} ((\Lambda \cap V_0)\oplus (\Lambda \cap V_0 \sqrt{-1}))
\]
is called a \textit{real lattice}. A \textit{real polarizable lattice} is a real lattice \((V_0,\Lambda)\) such that there exists a positive definite symmetric bilinear form \(S\) on \(V_0\) satisfying the condition that
\[
S(y_1,x_2) - S(x_1,y_2) \in \mathbb{Z}
\]
for every \(\{x_1,x_2,y_1,y_2\} \subset V_0\) such that \(x_1+y_1 \sqrt{-1}, x_2+y_2\sqrt{-1} \in \Lambda\). In this terminology, the two main results of the article are the following theorems.
{Theorem 1.} The isomorphism classes of real abelian varieties are in one-to-one correspondence with the isomorphism classes of real polarizable lattices.
{Theorem 2.} The Nash equivalence classes of abelian Nash manifolds are in one-to-one correspondence with the equivalence classes of real polarizable lattices modulo imaginary isogenies.
Note: In a follow-up paper, \textit{Y. Bao} et al. [J. Pure Appl. Algebra 226, No. 5, Article ID 106942, 19 p. (2022; Zbl 1490.14078)] extended Theorem 2 to the setting of the commutative locally Nash groups.
Reviewer: Can Mahir Bilen (New Orleans)Finite torsors over strongly \(F\)-regular singularitieshttps://zbmath.org/1491.140672022-09-13T20:28:31.338867Z"Carvajal-Rojas, Javier"https://zbmath.org/authors/?q=ai:carvajal-rojas.javierSummary: We investigate finite torsors over big opens of spectra of strongly \(F\)-regular germs that do not extend to torsors over the whole spectrum. Let \((R,\mathfrak{m},\mathfrak{k},K)\) be a strongly \(F\)-regular \(\mathfrak{k}\)-germ where \(\mathfrak{k}\) is an algebraically closed field of characteristic \(p>0\). We prove the existence of a finite local cover \(R\subset R^*\) so that \(R^*\) is a strongly \(F\)-regular \(\mathfrak{k}\)-germ and: for all finite algebraic groups \(G/\mathfrak{k}\) with solvable neutral component, every \(G\)-torsor over a big open of \(\mathrm{Spec}R^*\) extends to a \(G\)-torsor everywhere. To achieve this, we obtain a generalized transformation rule for the \(F\)-signature under finite local extensions. Such formula is used to show that the torsion of \(\mathrm{Cl}R\) is bounded by \(1/s(R)\). By taking cones, we conclude that the Picard group of globally \(F\)-regular varieties is torsion-free. Likewise, this shows that canonical covers of \(\mathbb{Q}\)-Gorenstein strongly \(F\)-regular singularities are strongly \(F\)-regular.Ext-groups in the category of strict polynomial functorshttps://zbmath.org/1491.140682022-09-13T20:28:31.338867Z"Tuan Pham, Van"https://zbmath.org/authors/?q=ai:tuan-pham.vanSummary: ``The aim of this paper is to study, by using the mathematical tools developed by Chałupnik, Touzé, and van der Kallen, the effect of the Frobenius twist on \(\mathrm{Ext}\)-group in the category of strict polynomial functors. As an application, we obtain explicit formulas of cohomology of the orthogonal groups and symplectic ones.''
The author gives a review of key results in the theory of strict polynomial functors. Along the way it is shown that they remain valid in the setting of strict polynomial functors in \(n\) variables. As an application a cohomological extension is then given, in a stable range, of the first and second fundamental theorems of invariant theory for the algebraic groups \(G_n=O_{n,n}\) and \(G_n=Sp_{n}\) over a base field \(\Bbbk\) of characteristic \(p>2\):
Let \(\Bbbk^{2n\vee(r)\oplus\ell}\) be the direct sum of \(\ell\) copies of the \(r\)-th Frobenius twist of the dual \(\Bbbk^{2n\vee}\) of the standard representation \(\Bbbk^{2n}\) of \(G_n\). If \(n\geq p^r\ell\) then
\[
H^*_\mathrm{rat}(G_n,S^*(\Bbbk^{2n\vee(r)\oplus\ell}))
\]
is a symmetric algebra on a finite set of generators
\[
(h|i|j)_{G_n}\in H^{2h}_\mathrm{rat}(G_n,S^*(\Bbbk^{2n\vee(r)\oplus\ell})),
\]
where \(0\leq h<p^r\), \(0\leq i\leq j \leq \ell\), and \(i\neq j\) if \(G_n\) is the symplectic group \(\mathrm{Sp_n}\). Here \(H^*_\mathrm{rat}\) refers to `rational cohomology', as appropriate for algebraic groups, and there are no relations among the \((h|i|j)_{G_n}\).
Note that for \(r=0\) the \(h=0\) part gives the first/second fundamental theorem as in [\textit {C. de Concini} and \textit{C. Procesi}, Adv. Math. 21, 330--354 (1976; Zbl 0347.20025)].
Reviewer: Wilberd van der Kallen (Utrecht)The motivic Satake equivalencehttps://zbmath.org/1491.140692022-09-13T20:28:31.338867Z"Richarz, Timo"https://zbmath.org/authors/?q=ai:richarz.timo"Scholbach, Jakob"https://zbmath.org/authors/?q=ai:scholbach.jakobSummary: We refine the geometric Satake equivalence due to
[\textit{V. Ginzburg}, ``Perverse sheaves on a Loop group and Langlands' duality'', Preprint, \url{arXiv:alg-geom/9511007};
\textit{A. Beilinson} and \textit{V. Drinfeld}, Quantization of Hitchin's integrable system and Hecke eigensheaves, Techn. Rep., MIT, 386 p. (1999);
\textit{I. Mirković} and \textit{K. Vilonen}, Ann. Math. (2) 166, No. 1, 95--143 (2007; Zbl 1138.22013)]
to an equivalence between mixed Tate motives on the double quotient \(L^+ G \setminus LG / L^+ G\) and representations of Deligne's modification of the Langlands dual group \({\widehat{G}} \).Products of elliptic curves and abelian surfaces by finite groups of automorphismshttps://zbmath.org/1491.140702022-09-13T20:28:31.338867Z"Hayashi, Taro"https://zbmath.org/authors/?q=ai:hayashi.taro.1|hayashi.taroSummary: In this paper, we work over \({{\mathbb{C}}} \). Criterions for an abelian surface to be products of elliptic curves are known. We give a new criterion from the view point of the Galois cover.On Schubert varieties of complexity onehttps://zbmath.org/1491.140712022-09-13T20:28:31.338867Z"Lee, Eunjeong"https://zbmath.org/authors/?q=ai:lee.eunjeong"Masuda, Mikiya"https://zbmath.org/authors/?q=ai:masuda.mikiya"Park, Seonjeong"https://zbmath.org/authors/?q=ai:park.seonjeongFix a Borel subgroup \(B\subseteq \text{GL}_n({\mathbb C})\), a maximal torus \(T\subseteq B\) and Weyl group \(W=S_n\) (the symmetric group). For the natural actions of \(T\) and \(B\) on the full flag variety \(\text{GL}_n({\mathbb C})/B\), every Schubert variety \(X_w\) is a \(T\)-invariant irreducible subvariety of \(\text{GL}_n({\mathbb C})/B\) and \(X_w\) is of \textit{complexity} \(k\) if a maximal \(T\)-orbit in \(X_w\) has codimension \(k\). For \(k=0\), that is, for toric Schubert varieties, in [\textit{P. Karuppuchamy}, Commun. Algebra 41, No. 4, 1365--1368 (2013; Zbl 1277.14039)] it is proved that \(X(w)\) is a toric variety if and only if a reduced expression for \(w\) has distinct letters and in [\textit{C. K. Fan}, Transform. Groups 3, No. 1, 51--56 (1998; Zbl 0912.20033)] there is a characterization of the elements \(w\in W\) whose reduced expressions avoid certain substrings and a criterion of smoothness of the corresponding Schubert variety. Moreover, in these cases \(X_w\) is isomorphic to a Bott-Samelson variety. \textit{B. E. Tenner} in [Adv. Appl. Math. 49, No. 1, 1--14 (2012; Zbl 1244.05010)] proves that \(w\) has a reduced decomposition with distinct letters if and only if \(w\) avoids the patterns \(321\) and \(3412\). Moreover, in [\textit{B. E. Tenner}, J. Comb. Theory, Ser. A 114, No. 5, 888-905 (2007; Zbl 1146.05054)] it is shown that a reduced decomposition of \(w\) consists of distinct letters if and only if the Bruhat interval \([e, w]\) is isomorphic to the Boolean algebra of rank the length of \(w\). A further characterization in terms of the Bruhat interval polytope was obtained by the authors in [\textit{E. Lee} et al., J. Comb. Theory, Ser. A 179, Article ID 105387, 42 p. (2021; Zbl 1467.52017)].
The main results of the paper under review are analog geometrical and combinatorial characterizations of Schubert varieties of complexity one. In this case, the characterizations are given in two theorems, one for the case when \(X_w\) is smooth and the other one when \(X_w\) is singular. In the former case, the authors use [\textit{V. Lakshmibai} and \textit{B. Sandhya}, Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45--52 (1990; Zbl 0714.14033)] characterization of smoothness of Schubert varieties in terms of pattern avoidance, namely: \(X_w\) is smooth if and only if \(w\) avoids the patterns the patterns \(3412\) and \(4321\). From here, the authors prove that \(X_w\) is smooth of complexity one if and only if \(w\) has the pattern \(321\) exactly once and avoids the pattern \(3412\). The corresponding result in the singular case becomes: \(X_w\) is singular of complexity one if and only if \(w\) has the pattern \(3412\) exactly once and avoids the pattern \(321\). Furthermore, using [\textit{D. Daly}, Graphs Comb. 29, No. 2, 173--185 (2013; Zbl 1263.05119)] characterization of certain reduced decompositions, the authors show that \(X_w\) is smooth of complexity one if and only if \(w\) has a reduced decomposition containing \(s_is_{i+1}s_i\) as a factor and no other repetitions, with a similar characterization for the singular case where now \(w\) having a reduced decomposition containing \(s_{i+1}s_is_{i+2}s_{i+1}\) as a factor and no other repetitions. Similarly, but with more involved details, the authors obtain characterizations of Schubert varieties of complexity one, for the smooth or singular case, in terms of generalized Bott-Samelson varieties or in terms of the Bruhat interval and polytope.
Reviewer: Felipe Zaldívar (Ciudad de México)Tropical geometry and Newton-Okounkov cones for Grassmannian of planes from compactificationshttps://zbmath.org/1491.140722022-09-13T20:28:31.338867Z"Manon, Christopher"https://zbmath.org/authors/?q=ai:manon.christopher"Yang, Jihyeon Jessie"https://zbmath.org/authors/?q=ai:yang.jihyeon-jessieFor the Grassmannian \(\text{Gr}_2({\mathbb C}^n)\) of two-dimensional vector subspaces of \({\mathbb C}^n\), embedded in the projective space \({\mathbb P}(\wedge^2{\mathbb C}^n)\simeq {\mathbb P}^{\binom{n}{2}-1}\) via the Plücker map, with vanishing ideal \(I_{2,n}\), let \(X\subseteq {\mathbb A}^{\binom{n}{2}}\) be the corresponding affine cone defined by the Plücker polynomials in \(I_{2,n}\). The (monomially free) initial ideals of \(I_{2,n}\) are parametrized by the tropical Grassmannian \(\text{Trop}(\text{Gr}_2({\mathbb C}^n))\) and in [Adv. Geom. 4, No. 3, 389--411 (2004; Zbl 1065.14071)] \textit{D. Speyer} and \textit{B. Sturmfels} showed that the maximal cones of \(\text{Trop}(\text{Gr}_2({\mathbb C}^n))\) are parametrized by labelled trivalent trees on \(n\) leaves, and the initial ideal associated with each of these cones is prime and binomial.
The main result of the paper under review is a different construction of this family of toric degenerations of the Grassmannian \(\text{Gr}_2({\mathbb C}^n)\). To do this, for each trivalent tree \(\sigma\) with \(n\) labeled leaves the authors obtain a compactification \(X_{\sigma}\) of \(X\) using a simplicial cone \(C_{\sigma}\) of rank \(1\) discrete valuations on \({\mathbb C}[X]\), one for each edge of the tree \(\sigma\), with common Khovanskii basis given by the Plücker generators and a rank \(2n-3\) discrete valuation \(v_{\sigma}\) with Khovanskii basis given by the Plücker generators. The compactification \(X_{\sigma}\supseteq X\) is obtained from a (combinatorial) normal crossing divisor \(D_{\sigma}\) that satisfies that the cone \(C_{\sigma}\) is spanned by the valuations associated to the components of the divisor and \(v_{\sigma}\) is a Parshin point valuation obtained from a flag of subvarieties of \(X_{\sigma}\) given by intersection components of the divisor. Furthermore, they show that the tropical variety \(\text{Trop}(\text{Gr}_2({\mathbb C}^n))\) can be recovered from the compactifications \(X_{\sigma}\) by proving that the affine semigroup algebra \({\mathbb C}[S_{\sigma}]\) associated to the value semigroup \(S_{\sigma}\) of the valuation \(v_{\sigma}\) has a presentation by the initial ideal of the cone in \(\text{Trop}(\text{Gr}_2({\mathbb C}^n))\) associated to \(\sigma\).
Lastly, using a choice of direct graph structure of the tree \(\sigma\) they show how to view \(X\) as certain type of quiver variety giving a description of each compactification \(X_{\sigma}\) in terms of the geometry of \(X\), and as by-product they show that each \(X_{\sigma}\) is a Fano variety.
Reviewer: Felipe Zaldívar (Ciudad de México)Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphismhttps://zbmath.org/1491.140732022-09-13T20:28:31.338867Z"Matsuzawa, Yohsuke"https://zbmath.org/authors/?q=ai:matsuzawa.yohsuke"Yoshikawa, Shou"https://zbmath.org/authors/?q=ai:yoshikawa.shouLet \(X\) be projective manifold defined over an algebraically closed field of characteristic zero. An endomorphism is a finite map \(f: X \rightarrow X\) of degree at least two. The (first) dynamical degree \(\delta_f\) measures the complexity of the dynamical system defined by \(f\), it can be computed as the spectral radius of the linear map \(f^* : N^1(X) \rightarrow N^1(X)\). If the manifold \(X\) is defined over \(\overline{\mathbb Q}\) and \(x \in X(\overline{\mathbb Q})\) is a point, the arithmetic degree \(\alpha_f(x)\) measures the arithmetic complexity of the orbit via some Weil height function [\textit{J. H. Silverman}, Ergodic Theory Dyn. Syst. 34, No. 2, 647--678 (2014; Zbl 1372.37093)]. \textit{S. Kawaguchi} and \textit{J. H. Silverman} [Trans. Am. Math. Soc. 368, No. 7, 5009--5035 (2016; Zbl 1391.37078)] have shown that the arithmetic degree is equal to the absolute value of one of the eigenvalues of \(f^*\), so one has an inequality \(\alpha_f(x) \leq \delta_f\). They also conjectured that if the \(f\)-orbit of the point is Zariski dense, one has equality \(\alpha_f(x) = \delta_f\). In this paper the authors prove the Kawaguchi-Silverman conjecture for rationally connected projective manifolds under the assumption that the endomorphism \(f\) is int-amplified, i.e. there exists an ample divisor \(H\) such that \(f^* H - H\) is ample.
From the point of view of the classification of projective varieties admitting an endomorphism, rationally connected manifolds are the most relevant case for this problem. The assumption that the endomorphism is int-amplified is more restrictive: it implies that the manifold \(X\) has only finitely many \(K_X\)-negative extremal rays and assures that one can run an MMP that is equivariant with respect to \(f\). In [``Kawaguchi-Silverman conjecture for surjective endomorphisms'', preprint, \url{arXiv:1908.01605}] this allowed \textit{S. Meng} and \textit{D.-Q. Zhang} to prove the Kawaguchi-Silverman conjecture for threefolds with klt singularities, they also proved the conjecture in any dimension under the assumption that certain Mori fibre spaces do not appear in the \(f\)-equivariant MMP.
In this paper, the authors are able to remove the assumption of Meng and Zhang by setting up a sophisticated argument by induction on the dimension and the Picard number of the variety. A crucial point is to prove that for certain Mori fibre spaces \(X \rightarrow Y\) in the MMP the base is a \(Q\)-abelian variety (i.e. a quotient \(A \rightarrow Y\) of an abelian variety \(A\) that does not ramify in codimension one) and that the quotient map \(A \rightarrow Y\) induces a map from an étale cover of \(X\) to an abelian variety. Since \(X\) is assumed to be smooth and rationally connected, such a map can not exist.
Reviewer: Andreas Höring (Nice)Staged tree models with toric structurehttps://zbmath.org/1491.140742022-09-13T20:28:31.338867Z"Görgen, Christiane"https://zbmath.org/authors/?q=ai:gorgen.christiane"Maraj, Aida"https://zbmath.org/authors/?q=ai:maraj.aida"Nicklasson, Lisa"https://zbmath.org/authors/?q=ai:nicklasson.lisaA staged tree model is a statistical model defined on a rooted tree whose edges are labeled by reals representing conditional probabilities, with the condition that the sum of labels of the edges starting from any vertex is \(1\). One can associate to such models \(T\) a real semialgebraic variety \(M_T\) in \(\mathbb R^q\), where \(q\) is the number of maximal paths of the tree, defined by the map which sends a choice of labels to the products of the labels along maximal paths. The main target of the authors is to study toric structures over \(M_T\), i.e. structures whose ideals are generated by binomials. When the tree is balanced, then the ideal of \(M_T\) is directly generated by binomials. The authors find conditions, which are satisfied in many non-balanced cases, for which there exists a determinantal ideal \(J\) which is generated by binomials and defines the variety \(M_T\).
Reviewer: Luca Chiantini (Siena)Parametrization, structure and Bruhat order of certain spherical quotientshttps://zbmath.org/1491.140752022-09-13T20:28:31.338867Z"Chaput, Pierre-Emmanuel"https://zbmath.org/authors/?q=ai:chaput.pierre-emmanuel"Fresse, Lucas"https://zbmath.org/authors/?q=ai:fresse.lucas"Gobet, Thomas"https://zbmath.org/authors/?q=ai:gobet.thomasLet \(G\) be a connected reductive algebraic group over an algebraically closed field of characteristic zero, let \(B\subset G\) be a Borel subgroup and let \(H=Z_G(e)\) be the isotropy group of a spherical nilpotent orbit \(G.e\subset\mathfrak g\). Recall that a nilpotent orbit \(G.e\) is spherical if and only if \(e\) is a nilpotent element of height less or equal to \(3\). Since the orbit \(G.e\) is here assumed to be spherical, it contains only finitely many \(B\)-orbits as well as the flag variety \(G/B\) has only finitely many \(H\)-orbits.
The authors study the structure of the isotropy group \(H\) and give a parameterization of the \(H\)-orbits in \(G/B\), provided \(e\) is a nilpotent element of height 2, showing that the \(H\)-orbits have the structure of algebraic affine bundles over certain homogeneous \(M\)-spaces, where \(M\) is a Levi subgroup of \(H\). Using their parameterization they introduce a partial order among the \(H\)-orbits and show that in type \(A\) their order coincides with the strong Bruhat order.
Reviewer: Paolo Bravi (Roma)Curves contracted by the Gauss maphttps://zbmath.org/1491.140762022-09-13T20:28:31.338867Z"Song, Lei"https://zbmath.org/authors/?q=ai:song.lei|song.lei.1Summary: Given a singular projective variety in some projective space, we characterize the smooth curves contracted by the Gauss map in terms of normal bundles. As a consequence, we show that if the variety is not linear, then a contracted line always has local obstruction for the embedded deformation and each component of the Hilbert scheme where the line lies is non-reduced everywhere.Stable pairs and Gopakumar-Vafa type invariants for Calabi-Yau \(4\)-foldshttps://zbmath.org/1491.140772022-09-13T20:28:31.338867Z"Cao, Yalong"https://zbmath.org/authors/?q=ai:cao.yalong"Maulik, Davesh"https://zbmath.org/authors/?q=ai:maulik.davesh"Toda, Yukinobu"https://zbmath.org/authors/?q=ai:toda.yukinobuThis paper studies the relationship between two types of enumerative invariants of Calabi-Yau \(4\)-folds \(X\). Pandharipande-Thomas (PT) invariants count pairs \(s\colon \mathcal{O}_X \to \mathcal{E}\), consisting of a pure \(1\)-dimensional coherent sheaf \(\mathcal{E}\) on \(X\) and a section \(s\), satisfying a certain stability condition. On the other hand, Gopakumar-Vafa (GV) invariants are conjecturally-integer quantities underlying the Gromov-Witten (GW) theory of \(X\), defined formally from its multiple-cover formulas. Earlier work [\textit{R. Pandharipande} and \textit{R. P. Thomas}, J. Am. Math. Soc. 23, No. 1, 267--297 (2010; Zbl 1250.14035)] proposed a relationship between PT and GV invariants for CY3s. The recent construction in [\textit{D. Borisov} and \textit{D. Joyce}, Geom. Topol. 21, No. 6, 3231--3311 (2017; Zbl 1390.14008)] of virtual fundamental classes \([M]^{\mathrm{vir}}\) for moduli spaces \(M\) of sheaves on CY4s enables the generalization of such a PT/GV correspondence to CY4s, modulo the (new) problem of how to consistently pick certain required orientation data for \(M\).
The main conjecture of this paper is that, for a suitable choice of orientation data,
\begin{align*}
P_{1,\beta}(\gamma_1, \ldots, \gamma_m) &= \sum_{\substack{\beta_1+\beta_2=\beta\\
\beta_1,\beta_2\ge 0}} n_{0,\beta_1}(\gamma_1, \ldots, \gamma_m) \cdot P_{0,\beta_2} \\
\sum_{\beta \ge 0} P_{0,\beta} q^\beta &= \prod_{\beta > 0} M(q^\beta)^{n_{1,\beta}}
\end{align*}
where the \(P_{n,\beta}\) (resp. \(n_{g,\beta}\)) are PT invariants (resp. GV invariants) of \(X\) in class \(\beta\), and \(M(q) = \prod_{k > 0} (1-q^k)^{-k}\) is the MacMahon function. Note that when \(n= \chi(\mathcal{E})=1\), non-trivial descendent insertions \(\gamma_i \in H^*(X; \mathbb{Z})\) are required for \(P_{1,\beta}\) to be non-zero, and when genus \(g>1\), GW and therefore the GV invariants \(n_{g,\beta}\) vanish on \(4\)-folds.
Many independent pieces of evidence are given to support the conjecture. A general proof is given assuming all relevant families of curves in \(X\) deform with expected properties. Explicit computational checks, for certain curve classes \(\beta\), are also given for: (compact case) sextic hypersurfaces, elliptic fibrations over \(\mathbb{P}^3\), and products of elliptic curves and CY3s; (non-compact case) local Fano \(3\)-folds, and local \(\mathbb{P}^1\) and elliptic curves. Many of these computations proceed by reduction to known conjectures/results for \(3\)-folds (e.g. [\textit{Y. Cao}, Commun. Contemp. Math. 22, No. 7, Article ID 1950060, 25 p. (2020; Zbl 1452.14053)]) or genus-\(0\) GV for CY4s defined using \(4\)-fold Donaldson-Thomas invariants (e.g. [\textit{Y. Cao} et al., Adv. Math. 338, 41--92 (2018; Zbl 1408.14177)]). Such reductions, especially to moduli spaces of sheaves on \(X\) of lower dimension, often give non-trivial insights on the correct choice of orientation in the \(4\)-fold setting. Finally, invariants for local curves are computed in low degrees via equivariant localization.
Reviewer: Huaxin Liu (Oxford)Relations in the tautological ring of the moduli space of curveshttps://zbmath.org/1491.140782022-09-13T20:28:31.338867Z"Pandharipande, R."https://zbmath.org/authors/?q=ai:pandharipande.rahul"Pixton, A."https://zbmath.org/authors/?q=ai:pixton.aaronLet \(\mathcal{M}_g\) be the space of nonsingular, projective, genus \(g\) curves over \(\mathbb{C}\). A classical fundamental problem is the study of its (Chow) cohomology ring and in particular the structure of a certain subring, the \emph{tautological ring}. To set some notation, denote by \(\pi:\mathcal{C}_g\to \mathcal{M}_g\) the universal curve and by \(\omega_\pi\) the dualizing sheaf. Define the \emph{cotangent line class}
\[
\psi=c_1(\omega_\pi)\in A^1(\mathcal{C}_g, \mathbb{Q})
\]
and the \emph{kappa} classes
\[
\kappa_r=\pi_*(\psi^{r+1})\in A^r(\mathcal{M}_g).
\]
The tautological ring
\[
R^*(\mathcal{M}_g)\subset A^r(\mathcal{M}_g, \mathbb{Q})
\]
is the \(\mathbb{Q}\)-subalgebra generated by all the \(\kappa\) classes, in other words there is a presentation
\[
\mathbb{Q}[\kappa_1, \kappa_2, \dots]\to R^*(\mathcal{M}_g)\to 0.
\]
By Mumford's conjecture -- proved by \textit{I. Madsen} and \textit{M. Weiss} [Ann. Math. (2) 165, No. 3, 843--941 (2007; Zbl 1156.14021)] -- the kappa classes exhaust all the \emph{stable cohomology}
\[
\lim_{g\to \infty}H^*(\mathcal{M}_g, \mathbb{Q})=\mathbb{Q}[\kappa_1, \kappa_2, \dots].
\]
Taking the opposite perspective, for a fixed \(g\) one may ask which relations generate the tautological ring. Furthermore, from the enumerative point of view, a large body of cycle class computations on \(\mathcal{M}_g\) lie in the tautological ring, and appear naturally in various algebro-geometric contexts (such as Brill-Noether theory).
Faber-Zagier conjectured a remarkable set of relations among the kappa classes in the tautological ring \( R^*(\mathcal{M}_g)\). The main result of the authors is a proof of such relations -- under a suitable formulation of the latter -- by a geometric construction involving the virtual class of the moduli space of stable quotients.
The authors define the polynomial
\[
\gamma^{FZ}=\sum_\sigma\sum_{r=0}^\infty C_r^{FZ}(\sigma), \kappa_r t^r \mathbf{p}^\sigma,
\]
where \(\sigma\) is a partition, \(t,\mathbf{p} \) are (multi)variables and \(C_r^{FZ}(\sigma) \) are some given explicit constants. The main result of the authors is the Faber-Zagier relation in \(R^r(\mathcal{M}_g)\)
\[
[\exp(-\gamma^{FZ})]_{t^r\mathbf{p}^\sigma}=0,
\]
whenever \(g-1+|\sigma|< 3r\) and \(g\equiv r+|\sigma|+1 \bmod 2\). Here, \([-]_{t^r\mathbf{p}^\sigma} \) denotes the coefficient of \(t^r\mathbf{p}^\sigma\), which is a polynomial in the \(\kappa_r\). It has to be noticed that this result yields \emph{finitely} many relations, for every given genus \(g\) and codimension \(r\).
The main result is obtained via Graber-Pandharipande virtual localization on the moduli space of stable quotients, where the problem is addressed with a combination of techniques from Geometry and Combinatorics.
Reviewer: Sergej Monavari (Utrecht)A quantum Leray-Hirsch theorem for banded gerbeshttps://zbmath.org/1491.140792022-09-13T20:28:31.338867Z"Tang, Xiang"https://zbmath.org/authors/?q=ai:tang.xiang"Tseng, Hsian-Hua"https://zbmath.org/authors/?q=ai:tseng.hsian-huaThe relationship between global and local is an eternal theme of mathematics. The classical Künneth formula expresses the cohomology of a product space as a tensor product of the cohomologies of the direct factors. Furthermore, if \(\pi: E\rightarrow B\) is a fiber bundle with fiber \(F\), the celebrated Leray-Hirsch theorem states that the cohomology of fiber bundle \(E\) is equal to the tensor product of the cohomologies of the base and fiber, i.e. \(H^* (E)\cong H^*(B)\otimes H^*(F)\). With the development of Gromov-Witten theory, a natural question is to ask : What is quantum geometry picture for the Leray-Hirsch theorem?
The paper under review tries to answer this question and studies the quantum version of Leray-Hirsch theorem in the context of orbifold Gromov-Witten theory of a gerbe \(\mathcal{Y}\rightarrow \mathcal{B}\) banded by a finite group \(G\), where \(\mathcal{B}\) is a smooth proper (i.e. compact) Deligne-Mumford stack and a \(G\)-gerbe \(\mathcal{Y}\rightarrow \mathcal{B}\) can be regarded as a fiber bundle over complex orbifold \(\mathcal{B}\) with fiber \(BG\). The main result (i.e. quantum Leray-Hirsch theorem) states that the Gromov-Witten theory of \(\mathcal{Y}\) can be expressed in terms of the Gromov-Witten theory of base \(\mathcal{B}\) and the information of fiber \(BG\). The key point of the proof is to analyze the properties and degree for the pushforward map \(\pi: \overline{\mathcal{M}}_{g, n}(\mathcal{Y}, \beta)\rightarrow \overline{\mathcal{M}}_{g, n}(\mathcal{B}, \beta)\).
In conclusion, the paper under review presents an important structural decomposition theorem for a banded \(G\)-gerbe over complex orbifold \(\mathcal{B}\) which can be regarded as a quantum Leray-Hirsch style theorem. The combinatorial degree formula and virtual pushforward formula are impressive in the proof.
Reviewer: Xiaobin Li (Chengdu)Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappingshttps://zbmath.org/1491.140802022-09-13T20:28:31.338867Z"Đinh, Sĩ Tiệp"https://zbmath.org/authors/?q=ai:dinh.si-tiep"Jelonek, Zbigniew"https://zbmath.org/authors/?q=ai:jelonek.zbigniew"Phạm, Tiến Sơn"https://zbmath.org/authors/?q=ai:pham-tien-son.Summary: Given a closed semi-algebraic set \(X\subset\mathbb{R}^n\) and a continuous semi-algebraic mapping \(G: X\to\mathbb{R}^m\), it will be shown that there exists an open dense semi-algebraic subset \(\mathscr{U}\) of \(L(\mathbb{R}^n,\mathbb{R}^m)\), the space of all linear mappings from \(\mathbb{R}^n\) to \(\mathbb{R}^m\), such that for all \(F\in\mathscr{U}\), the image \((F+G)(X)\) is a closed (semi-algebraic) set in \(\mathbb{R}^m\). To do this, we study the tangent cone at infinity \(C_\infty X\) and the set \(E_\infty X\subset C_\infty X\) of (unit) exceptional directions at infinity of \(X\). Specifically we show that the set \(E_\infty X\) is nowhere dense in \(C_\infty X\cap\mathbb{S}^{n-1}\).Counting isolated points outside the image of a polynomial maphttps://zbmath.org/1491.140812022-09-13T20:28:31.338867Z"El Hilany, Boulos"https://zbmath.org/authors/?q=ai:el-hilany.boulosSummary: We consider a generic family of polynomial maps \(f := (f_1, f_2): \mathbb{C}^2 \rightarrow \mathbb{C}^2\) with given supports of polynomials, and degree deg \(f := \max(\operatorname{deg} f_1, \operatorname{deg} f_2)\). We show that the (non-) properness of maps \(f\) in this family depends uniquely on the pair of supports, and that the set of isolated points in \(\mathbb{C}^2 \setminus f (\mathbb{C}^2)\) has a size of at most 6 deg \(f\). This improves an existing upper bound \((\operatorname{deg} f - 1)^2\) proven by Jelonek. Moreover, for each \(n \in \mathbb{N} \), we construct a dominant map \(f\) as above, with \(\operatorname{deg} f = 2 n + 2\), and having \(2n\) isolated points in \(\mathbb{C}^2 \setminus f (\mathbb{C}^2)\). Our proofs are constructive and can be adapted to a method for computing isolated missing points of \(f\). As a byproduct, we describe those points in terms of singularities of the bifurcation set of \(f\).On finiteness theorems of polynomial functionshttps://zbmath.org/1491.140822022-09-13T20:28:31.338867Z"Koike, Satoshi"https://zbmath.org/authors/?q=ai:koike.satoshi"Paunescu, Laurentiu"https://zbmath.org/authors/?q=ai:paunescu.laurentiuSummary: Let \(d\) be a positive integer. We show a finiteness theorem for semialgebraic \(\mathscr{RL}\) triviality of a Nash family of Nash functions defined on a Nash manifold, generalising Benedetti-Shiota's finiteness theorem for semialgebraic \(\mathscr{RL}\) equivalence classes appearing in the space of real polynomial functions of degree not exceeding \(d\). We also prove Fukuda's claim, Theorem 1.3, and its semialgebraic version Theorem 1.4, on the finiteness of the local \({\mathscr{R}}\) types appearing in the space of real polynomial functions of degree not exceeding \(d\).Generalizations of Samuel's criteria for a ring to be a unique factorization domainhttps://zbmath.org/1491.140832022-09-13T20:28:31.338867Z"Daigle, Daniel"https://zbmath.org/authors/?q=ai:daigle.daniel"Freudenburg, Gene"https://zbmath.org/authors/?q=ai:freudenburg.gene"Nagamine, Takanori"https://zbmath.org/authors/?q=ai:nagamine.takanoriGiven a unique factorization domain (UFD), the authors study conditions on ring extensions to be UFDs as well. Their results include inter alia generalizations of criteria given by \textit{P. Samuel} [Lectures on unique factorization domains. Notes by M. Pavman Murthy. Bombay: Tata Institute of Fundamental Research (1965; Zbl 0184.06601)] and \textit{S. Mori} [Jpn. J. Math., New Ser. 3, 223--238 (1977; Zbl 0393.13003)]. As application they construct \(\mathbb{Z}\)-graded non-noetherian rational UFDs of dimension \(3\) over an arbitrary field \(k\). Moreover, they show that a certain class of rings defined by trinomial relations, which appear i.a. as Cox rings of varieties with torus ation of complexity one, are UFDs.
Reviewer: Milena Wrobel (Oldenburg)On the existence of \(B\)-root subgroups on affine spherical varietieshttps://zbmath.org/1491.140842022-09-13T20:28:31.338867Z"Avdeev, R. S."https://zbmath.org/authors/?q=ai:avdeev.roman-s"Zhgoon, V. S."https://zbmath.org/authors/?q=ai:zhgoon.vladimir-sSummary: Let \(X\) be an irreducible affine algebraic variety that is spherical with respect to an action of a connected reductive group \(G\). In this paper, we provide sufficient conditions, formulated in terms of weight combinatorics, for the existence of one-parameter additive actions on \(X\) normalized by a Borel subgroup \(B \subset G\). As an application, we prove that every \(G\)-stable prime divisor in \(X\) can be connected with an open \(G\)-orbit by means of a suitable \(B\)-normalized one-parameter additive action.
See also [\textit{I. Arzhantsev} and the first author, Sel. Math., New Ser. 28, No. 3, Paper No. 60, 37 p. (2022; Zbl 07523717)].Mirror symmetry on levels of non-abelian Landau-Ginzburg orbifoldshttps://zbmath.org/1491.140852022-09-13T20:28:31.338867Z"Ebeling, Wolfgang"https://zbmath.org/authors/?q=ai:ebeling.wolfgang"Gusein-Zade, Sabir M."https://zbmath.org/authors/?q=ai:gusein-zade.sabir-mSummary: We consider the Berglund-Hübsch-Henningson-Takahashi duality of Landau-Ginzburg orbifolds with a symmetry group generated by some diagonal symmetries and some permutations of variables. We study the orbifold Euler characteristics, the orbifold monodromy zeta functions and the orbifold E-functions of such dual pairs. We conjecture that we get a mirror symmetry between these invariants even on each level, where we call level the conjugacy class of a permutation. We support this conjecture by giving partial results for each of these invariants.Computation of Dressians by dimensional reductionhttps://zbmath.org/1491.140862022-09-13T20:28:31.338867Z"Brandt, Madeline"https://zbmath.org/authors/?q=ai:brandt.madeline"Speyer, David E."https://zbmath.org/authors/?q=ai:speyer.david-eSummary: We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. We give counterexamples to a few plausible statements about matroid subdivisions.On the tropical discrete logarithm problem and security of a protocol based on tropical semidirect producthttps://zbmath.org/1491.150302022-09-13T20:28:31.338867Z"Muanalifah, Any"https://zbmath.org/authors/?q=ai:muanalifah.any"Sergeev, Sergeĭ"https://zbmath.org/authors/?q=ai:sergeev.sergei-m|sergeev.sergei-n|sergeev.sergei-alekseevichSummary: Tropical linear algebra has been recently put forward by \textit{D. Grigoriev} and \textit{V. Shpilrain} [Commun. Algebra 42, No. 6, 2624--2632 (2014; Zbl 1301.94114); Commun. Algebra 47, No. 10, 4224--4229 (2019; Zbl 1451.14179)] as a promising platform for implementation of protocols of Diffie-Hellman and Stickel type. Based on the CSR expansion of tropical matrix powers, we suggest a simple algorithm for the following tropical discrete logarithm problem: ``Given that \(A=V\otimes F^{\otimes t}\) for a unique \(t\) and matrices \(A, V, F\) of appropriate dimensions, find this \(t\).'' We then use this algorithm to suggest a simple attack on a protocol based on the tropical semidirect product. The algorithm and the attack are guaranteed to work in some important special cases and are shown to be efficient in our numerical experiments.Double framed moduli spaces of quiver representationshttps://zbmath.org/1491.160132022-09-13T20:28:31.338867Z"Armenta, Marco"https://zbmath.org/authors/?q=ai:armenta.marco-antonio"Brüstle, Thomas"https://zbmath.org/authors/?q=ai:brustle.thomas"Hassoun, Souheila"https://zbmath.org/authors/?q=ai:hassoun.souheila"Reineke, Markus"https://zbmath.org/authors/?q=ai:reineke.markusSummary: Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a network category whose isomorphism classes of objects correspond to the orbits of quiver representations, in which neural networks map input data. We then prove that the output of a neural network depends only on the corresponding point in the moduli space. Finally, we present a different perspective on mapping neural networks with a specific activation function, called ReLU, to a moduli space using the symplectic reduction approach to quiver moduli.Lattices, spectral spaces, and closure operations on idempotent semiringshttps://zbmath.org/1491.160462022-09-13T20:28:31.338867Z"Jun, Jaiung"https://zbmath.org/authors/?q=ai:jun.jaiung"Ray, Samarpita"https://zbmath.org/authors/?q=ai:ray.samarpita"Tolliver, Jeffrey"https://zbmath.org/authors/?q=ai:tolliver.jeffreySummary: Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the \textit{prime \(k\)-spectrum} of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the \textit{space of valuations} and the \textit{space of prime congruences} on an idempotent semiring are spectral, and there is a natural bijection of sets between the two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of \textit{closure operations} for semirings, and provide several examples. In particular, we introduce an \textit{integral closure operation} and a \textit{Frobenius closure operation} for idempotent semirings.Quasi-split symmetric pairs of \(U(\mathfrak{gl}_N)\) and their Schur algebrashttps://zbmath.org/1491.170072022-09-13T20:28:31.338867Z"Li, Yiqiang"https://zbmath.org/authors/?q=ai:li.yiqiang|li.yiqiang.1"Zhu, Jieru"https://zbmath.org/authors/?q=ai:zhu.jieruA classical result in representation theory states that the action of the symmetric group \(S_d\) fully centralizes the natural action of the complex general linear algebra \(\mathfrak{gl}_N\) on the tensor space \((\mathbb C^N)^{\otimes d}\). This results in representations for \(\mathfrak{gl}_N\), which are summands of \((\mathbb C^N)^{\otimes d}\), being in bijection with representations for the symmetric group \(S_d\). The Schur algebra of type A is the centralizer algebra of \(S_d\) on \((\mathbb{C}^N)^{\otimes d}\).
In type B, the orthogonal group does not fully centralize the action of the type B Weyl group on the tensor space. The orthogonal group is known to centralize the action of the Brauer algebra but the description of the centralizer of the type B Weyl group action, and its quantization, is nontrivial. In one approach, the centralizer is given by a subgroup of \(GL_N\), its Lie algebra being the fixed-point subalgebra \(\mathfrak{gl}_N^{\theta}\) of \(\mathfrak{gl}_N\) under a certain involution \(\theta\). In another approach, the centralizer is given as a homomorphic image of a two-block subalgebra \(U\) of \(U(\mathfrak{gl}_N)\). The pairs \((U(\mathfrak{gl}_N), U)\) and \((U(\mathfrak{gl}_N), U(\mathfrak{gl}_N^{\theta}))\) are infinitesimal quasi-split symmetric pairs of type A.
Y. Li and J. Zhu establish explicit isomorphisms between \(U\) and \(U(\mathfrak{gl}_N^{\theta})\) (Theorem 2.4.2, page 14), and consequently on their respective Schur algebras. The authors also provide a presentation of the geometric counterpart of the above Schur algebras specialized at \(q=1\).
Reviewer: Mee Seong Im (Annapolis)Bracket width of simple Lie algebrashttps://zbmath.org/1491.170172022-09-13T20:28:31.338867Z"Dubouloz, Adrien"https://zbmath.org/authors/?q=ai:dubouloz.adrien"Kunyavskiĭ, Boris"https://zbmath.org/authors/?q=ai:kunyavskii.boris-e"Regeta, Andriy"https://zbmath.org/authors/?q=ai:regeta.andriyConsider a simple Lie algebra \(L\). Then \([L,L]=L\) (note that for associative \(PI\)-algebra \(A\) equality \([A,A]=A\) never holds [\textit{A. Ya. Belov}, Sib. Mat. Zh. 44, No. 6, 1239--1254 (2003; Zbl 1054.16015); translation in Sib. Math. J. 44, No. 6, 969--980 (2003)]. In analogy to the group case, the authors deal with notion of {\em commutator with} i.e. minimal \(n\) such that any element \(x\in L\) can be presented as a sum \[x=H_n=\sum_{i=1}^n [a_i,b_i].\]
This can be viewed as a question of possible values of polynomial \(H_n\) on the algebra \(L\) (see review of [the reviewer et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 16, Paper 071, 61 p. (2020; Zbl 1459.16012)])
For finite dimensional simple Lie algebras over algebraically closed field of characteristic zero commutator width equal 1, for many other fields including reals the answer is still unknown.
The paper deals with Lie algebras of (sometimes symplectic) vector fields on algebraic varieties mostly with curves and Danilevsky surfaces. Algebraic geometry properties of these varieties cause some width consequences. Note that Yu.P. Razmyslov established restitution of algebraic variety \(M\) by the Lie algebra of vector fields on \(M\) (I recommend the very deep book [\textit{Yu. P. Razmyslov}, Identities of algebras and their representations. Providence, RI: American Mathematical Society (1994; Zbl 0827.17001)]) In non-rational case for curves, the authors proved that commutator width is at least \(2\). For Danilevsky surfaces some relations with Jacobian Conjecture type questions are established.
The authors discuss some open problems including possible commutator width. Can it be arbitrary large for vector field algebras on algebraic varieties?
Reviewer: Alexei Kanel-Belov (Ramat-Gan)The algebraic and geometric classification of nilpotent right commutative algebrashttps://zbmath.org/1491.170212022-09-13T20:28:31.338867Z"Adashev, Jobir"https://zbmath.org/authors/?q=ai:adashev.jobir-q"Kaygorodov, Ivan"https://zbmath.org/authors/?q=ai:kaigorodov.i-b"Khudoyberdiyev, Abror"https://zbmath.org/authors/?q=ai:khudoyberdiyev.abror-kh"Sattarov, Aloberdi"https://zbmath.org/authors/?q=ai:sattarov.aloberdiIn this paper, the authors give a complete algebraic and geometric classification of complex 4-dimensional nilpotent right commutative algebras. In Theorem A, they prove that a complex 4-dimensional nilpotent right commutative algebra is a Novikov algebra or it is isomorphic to an algebra A, with the classification given in a table which includes all cases. They prove that the corresponding geometric variety has dimension 15 and has a decomposition in 5 irreducible components determined by the Zariski closures of four one-parameter families of algebras and a two-parameter family of algebras. The result has been done in Theorem B, followed by a table in which all cases are presented.
Reviewer: Cristina Flaut (Constanta)On an analogue of \(L^2\)-Betti numbers for finite field coefficients and a question of Atiyahhttps://zbmath.org/1491.200022022-09-13T20:28:31.338867Z"Neumann, Johannes"https://zbmath.org/authors/?q=ai:neumann.johannesSummary: We construct a dimension function for modules over the group ring of an amenable group. This may replace the von Neumann dimension in the definition of \(L^2\)-Betti numbers and thus allows an analogous definition for finite field coefficients. Furthermore we construct examples for characteristic 2 in answer to Atiyah question of irrational \(L^2\)-Betti numbers.On the moduli spaces of commuting elements in the projective unitary groupshttps://zbmath.org/1491.200232022-09-13T20:28:31.338867Z"Adem, Alejandro"https://zbmath.org/authors/?q=ai:adem.alejandro"Cheng, Man Chuen"https://zbmath.org/authors/?q=ai:cheng.man-chuenSummary: We provide descriptions for the moduli spaces $\mathrm{Rep}(\Gamma, PU(m))$, where $\Gamma$ is any finitely generated abelian group and $PU(m)$ is the group of \(m \times m\) projective unitary matrices. As an application, we show that for any connected CW-complex \(X\) with $\pi_1(X) \cong \mathbb{Z}^n$, the natural map $\pi_{0}(\mathrm{Rep}(\pi_{1}(X), PU(m))) \rightarrow [X, BPU(m)]$ is injective, hence providing a complete enumeration of the isomorphism classes of flat principal $PU(m)$-bundles over \(X\).
{\par\copyright 2019 American Institute of Physics}Uniqueness of polarization for the autonomous 4-dimensional Painlevé-type systemshttps://zbmath.org/1491.340992022-09-13T20:28:31.338867Z"Nakamura, Akane"https://zbmath.org/authors/?q=ai:nakamura.akane"Rains, Eric"https://zbmath.org/authors/?q=ai:rains.eric-mThe paper discusses an autonomous 4-dimensional integral system of Painlevé type. The study of such system is noticeably simplified if it is possible to build the so called Lax pair for it. In this paper, the authors present a systematic way to construct such Lax pairs. The key statement of the paper is the following one.
\textbf{Theorem 1.} For the 4-dimensional autonomous Painlevé-type equations, the Jacobian of the generic spectral curve has no nontrivial endomorphism.
Reviewer: Mykola Grygorenko (Kyïv)Geometry and dynamics on Riemann and \(K3\) surfaceshttps://zbmath.org/1491.370602022-09-13T20:28:31.338867Z"Filip, Simion"https://zbmath.org/authors/?q=ai:filip.simionSummary: Surfaces are some of the simplest yet geometrically rich manifolds. Geometric structures on surfaces illuminate their topology and are useful for studying dynamical systems on surfaces. We illustrate below how some of these concepts blend together, and relate them to algebraic geometry.The holonomy of a singular leafhttps://zbmath.org/1491.530312022-09-13T20:28:31.338867Z"Laurent-Gengoux, Camille"https://zbmath.org/authors/?q=ai:laurent-gengoux.camille"Ryvkin, Leonid"https://zbmath.org/authors/?q=ai:ryvkin.leonid\textit{I. Androulidakis} and \textit{G. Skandalis} [J. Reine Angew. Math. 626, 1--37 (2009; Zbl 1161.53020)] gave a construction for the holonomy groupoid of any singular foliation. Androulidakis and Zambon formulated a holonomy map for singular foliations, which is defined on the holonomy groupoid, rather than the fundamental group of a leaf, as it happens with regular foliations. On the other hand, Laurent-Gengoux, Lavau and Strobl established a universal Lie-\(\infty\) algebroid to every singular foliation.
In the paper under review, the authors construct higher holonomy maps, defined on \(\pi_n(L)\), where \(L\) is a singular leaf \(L\). They take values in the \((n-1)\)-th homotopy group of the universal Lie-\(\infty\) algebroid associated with the transversal foliation to \(L\). Moreover, they show that these holonomy maps form a long exact sequence.
Reviewer: Iakovos Androulidakis (Athína)Heavenly metrics, BPS indices and twistorshttps://zbmath.org/1491.530632022-09-13T20:28:31.338867Z"Alexandrov, Sergei"https://zbmath.org/authors/?q=ai:alexandrov.sergei-yu"Pioline, Boris"https://zbmath.org/authors/?q=ai:pioline.borisSummary: Recently, \textit{T. Bridgeland} [``Geometry from Donaldson-Thomas invariants'', Preprint, \url{arXiv:1912.06504}] defined a complex hyperkähler metric on the tangent bundle over the space of stability conditions of a triangulated category, based on a Riemann-Hilbert problem determined by the Donaldson-Thomas invariants. This metric is encoded in a function \(W(z,\theta)\) satisfying a heavenly equation, or a potential \(F(z,\theta)\) satisfying an isomonodromy equation. After recasting the RH problem into a system of TBA-type equations, we obtain integral expressions for both \(W\) and \(F\) in terms of solutions of that system. These expressions are recognized as conformal limits of the `instanton generating potential' and `contact potential' appearing in studies of D-instantons and BPS black holes. By solving the TBA equations iteratively, we reproduce Joyce's original construction of \(F\) as a formal series in the rational DT invariants. Furthermore, we produce similar solutions to deformed versions of the heavenly and isomonodromy equations involving a non-commutative star product. In the case of a finite uncoupled BPS structure, we rederive the results previously obtained by Bridgeland and obtain the so-called \(\tau\) function for arbitrary values of the fiber coordinates \(\theta\), in terms of a suitable two-variable generalization of Barnes' \(G\) function.A note on Griffiths' conjecture about the positivity of Chern-Weil formshttps://zbmath.org/1491.530782022-09-13T20:28:31.338867Z"Fagioli, Filippo"https://zbmath.org/authors/?q=ai:fagioli.filippoLet \((E, h)\) be a Griffiths semipositive Hermitian holomorphic vector bundle of rank 3 over a complex manifold. In this paper, the author proves the positivity of the characteristic differential form
\[
c_1(E, h)\wedge c_2(E, h) -c_3(E, h).
\]
Here, for \(k=1,2,3\), \(c_k(E,h)\) denotes the Chern form of bidegree \((k,k)\), which represents the Chern class \(c_k(E)\) of the vector bundle \(E\). This provides a new evidence towards Griffiths' conjecture about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, the author establishes the following new chain of inequalities between Chern forms
\[
c_1(E, h)^3 \ge c_1(E, h) \wedge c_2(E, h) \ge c_3(E, h).
\]
The author also shows how to obtain the positivity of the second Chern form \(c_2(E, h)\) in any rank, if \((E, h)\) is Griffiths positive. This is obtained by adapting the Griffiths' result on the positivity of \(c_2(E, h)\) in rank 2.
The final part of the paper gives an overview on the state of the art of Griffiths' conjecture, collecting several remarks and open questions.
Reviewer: Riccardo Piovani (Parma)Group bundles and group connectionshttps://zbmath.org/1491.580102022-09-13T20:28:31.338867Z"Blázquez-Sanz, D. B."https://zbmath.org/authors/?q=ai:blazquez-sanz.d-b"Marín Arango, C. A."https://zbmath.org/authors/?q=ai:marin-arango.carlos-alberto"Suárez Gordon, S."https://zbmath.org/authors/?q=ai:suarez-gordon.sSummary: We consider smooth families of Lie groups (group bundles) and connections that are compatible with the group operation. We characterize the space of group connections on a group bundle as an affine space modeled over the vector space of 1-forms with values cocycles in the Lie algebra bundle of the aforementioned group bundle. We show that group connections satisfy the Ambrose-Singer theorem and that group bundles can be seen as a particular case of associated bundles realizing group connections as associated connections. We give a construction of the moduli space of group connections with fixed base and fiber, as a space of representations of the fundamental group of the base.Pairs of Lie-type and large orbits of group actions on filtered modules: a characteristic-free approach to finite determinacyhttps://zbmath.org/1491.580132022-09-13T20:28:31.338867Z"Boix, Alberto F."https://zbmath.org/authors/?q=ai:boix.alberto-f"Greuel, Gert-Martin"https://zbmath.org/authors/?q=ai:greuel.gert-martin"Kerner, Dmitry"https://zbmath.org/authors/?q=ai:kerner.dmitrySummary: Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action and the first step is always to reduce the determinacy question to an ``infinitesimal determinacy'', i.e., to the tangent spaces at the orbits of the group action. In this work we formulate a universal, characteristic-free approach to finite determinacy, not necessarily over a field, and for a large class of group actions. We do not restrict to pro-algebraic or Lie groups, rather we introduce the notion of ``pairs of (weak) Lie type'', which are groups together with a substitute for the tangent space to the orbit such that the orbit is locally approximated by its tangent space, in a precise sense. This construction may be considered as a kind of replacement of the exponential resp. logarithmic maps. It is of independent interest as it provides a general method to pass from the tangent space to the orbit of a group action in any characteristic. In this generality we establish the ``determinacy versus infinitesimal determinacy'' criteria, a far reaching generalization of numerous classical and recent results, together with some new applications.FZZ-triality and large \(\mathcal{N} = 4\) super Liouville theoryhttps://zbmath.org/1491.810252022-09-13T20:28:31.338867Z"Creutzig, Thomas"https://zbmath.org/authors/?q=ai:creutzig.thomas"Hikida, Yasuaki"https://zbmath.org/authors/?q=ai:hikida.yasuakiSummary: We examine dualities of two dimensional conformal field theories by applying the methods developed by the authors. We first derive the duality between \(SL(2|1)_k/(SL(2)_k \otimes U(1))\) coset and Witten's cigar model or sine-Liouville theory. The latter two models are Fateev-Zamolodchikov-Zamolodchikov (FZZ-)dual to each other, hence the relation of the three models is named FZZ-triality. These results are used to study correlator correspondences between large \(\mathcal{N} = 4\) super Liouville theory and a coset of the form \(Y(k_1, k_2)/SL(2)_{k_1 + k_2}\), where \(Y(k_1, k_2)\) consists of two \(SL(2|1)_{k_i}\) and free bosons or equivalently two \(U(1)\) cosets of \(D(2, 1; k_i - 1)\) at level one. These correspondences are a main result of this paper. The FZZ-triality acts as a seed of the correspondence, which in particular implies a hidden \(SL(2)_{k^\prime}\) in \(SL(2|1)_k\) or \(D(2, 1; k - 1)_1\). The relation of levels is \(k^\prime - 1 = 1/(k-1)\). We also construct boundary actions in sine-Liouville theory as another use of the FZZ-triality. Furthermore, we generalize the FZZ-triality to the case with \(SL(n|1)_k /(SL(n)_k \otimes U(1))\) for arbitrary \(n > 2\).Pion form factor from an AdS deformed backgroundhttps://zbmath.org/1491.810292022-09-13T20:28:31.338867Z"Martín Contreras, Miguel Angel"https://zbmath.org/authors/?q=ai:martin-contreras.miguel-angel"Capossoli, Eduardo Folco"https://zbmath.org/authors/?q=ai:capossoli.eduardo-folco"Li, Danning"https://zbmath.org/authors/?q=ai:li.danning"Vega, Alfredo"https://zbmath.org/authors/?q=ai:vega.alfredo"Boschi-Filho, Henrique"https://zbmath.org/authors/?q=ai:boschi-filho.henriqueSummary: We consider a bottom-up AdS/QCD model with a conformal exponential deformation \(e^{k_I z^2}\) on a Lorentz invariant AdS background, where \(k_I\) stands for the scale \(k_\pi\) that fixes confinement in the pion case and \(k_\gamma\) for the kinematical energy scale associated with the virtual photon. In this model we assume the conformal dimension associated with the operator that creates pions at the boundary as \(\Delta = 3\), as in the original bottom-up AdS/QCD proposals. Regarding the geometric slope related to photon field \(k_\gamma\), we analyze two cases: constant and depending on the transferred momentum \(q\). In these two cases we computed the electromagnetic pion form factor as well as the pion radius. We compare our results with experimental data as well as other theoretical (holographic and non-holographic) models. In particular, for the momentum dependent scale, we find good agreement with the available experimental data as well as non-holographic models.Geometric dark matterhttps://zbmath.org/1491.830192022-09-13T20:28:31.338867Z"Demir, Durmuş"https://zbmath.org/authors/?q=ai:demir.durmus-ali"Puliçe, Beyhan"https://zbmath.org/authors/?q=ai:pulice.beyhan(no abstract)Topology of cosmological black holeshttps://zbmath.org/1491.830342022-09-13T20:28:31.338867Z"Mirbabayi, Mehrdad"https://zbmath.org/authors/?q=ai:mirbabayi.mehrdad(no abstract)Duality of sum of nonnegative circuit polynomials and optimal SONC boundshttps://zbmath.org/1491.901162022-09-13T20:28:31.338867Z"Papp, Dávid"https://zbmath.org/authors/?q=ai:papp.davidSummary: Circuit polynomials are polynomials with properties that make it easy to compute sharp and certifiable global lower bounds for them. Consequently, one may use them to find certifiable lower bounds for any polynomial by writing it as a sum of circuit polynomials with known lower bounds. Recent work has shown that sums of nonnegative circuit polynomials (or SONC polynomials for short) can be used to compute global lower bounds (called SONC bounds) for polynomials in this manner very efficiently both in theory and in practice, if the polynomial is bounded from below and its support satisfies a certain nondegeneracy assumption. The quality of the SONC bound depends on the circuits used in the computation but finding the set of circuits that yield the best attainable SONC bound among the astronomical number of candidate circuits is a non-trivial task that has not been addressed so far. We propose an efficient method to compute the optimal SONC lower bound by iteratively identifying the optimal circuits to use in the SONC bounding process. The method is derived from a new proof of the result that every SONC polynomial decomposes into SONC polynomials on the same support. This proof is based on convex programming duality and motivates a column generation approach that is particularly attractive for sparse polynomials of high degree and with many unknowns. The method is implemented and tested on a large set of sparse polynomial optimization problems with up to 40 unknowns, of degree up to 60, and up to 3000 monomials in the support. The results indicate that the method is efficient in practice and requires only a small number of iterations to identify the optimal circuits.Resistance of isogeny-based cryptographic implementations to a fault attackhttps://zbmath.org/1491.940672022-09-13T20:28:31.338867Z"Tasso, Élise"https://zbmath.org/authors/?q=ai:tasso.elise"De Feo, Luca"https://zbmath.org/authors/?q=ai:de-feo.luca"El Mrabet, Nadia"https://zbmath.org/authors/?q=ai:el-mrabet.nadia"Pontié, Simon"https://zbmath.org/authors/?q=ai:pontie.simonSummary: The threat of quantum computers has sparked the development of a new kind of cryptography to resist their attacks. Isogenies between elliptic curves are one of the tools used for such cryptosystems. They are championed by SIKE (Supersingular isogeny key encapsulation), an ``alternate candidate'' of the third round of the NIST Post-Quantum Cryptography Standardization Process. While all candidates are believed to be mathematically secure, their implementations may be vulnerable to hardware attacks. In this work we investigate for the first time whether Ti's theoretical fault injection attack [\textit{Y. B. Ti}, Lect. Notes Comput. Sci. 10346, 107--122 (2017; Zbl 1437.94075)] is exploitable in practice. We also examine suitable countermeasures. We manage to recover the secret thanks to electromagnetic fault injection on an ARM Cortex A53 using a correct and an altered public key generation. Moreover we propose a suitable countermeasure to detect faults that has a low overhead as it takes advantage of a redundancy already present in SIKE implementations.
For the entire collection see [Zbl 1489.94002].