×

Five embeddings of one simple group. (English) Zbl 1291.14060

The Cremona group \(\mathrm{Cr}_3({\mathbb{C}})\) is the group of birational automorphisms of the projective space \({\mathbb{P}}^3\). The authors investigate the following two questions and are able to answer some cases. The first question was raised by J.-P. Serre in 2010: What are normalizers in \(\mathrm{Cr}_3({\mathbb{C}})\) of finite simple non-abelian subgroups in \(\mathrm{Cr}_3({\mathbb{C}})\)? The second question is: How to decide whether or not two finite isomorphic subgroups in \(\mathrm{Cr}_3({\mathbb{C}})\) are conjugate?
The main result is Theorem 1.5.
{ Theorem 1.5.} Up to conjugation, there are at least 5 subgroups in \(\mathrm{Cr}_3({\mathbb{C}})\) that are isomorphic to \(A_6\). For three of these non-conjugated subgroups, the normalizer in \(\mathrm{Cr}_3({\mathbb{C}})\) is \(S_6\), and for one it is the free product of \(S_6\) and \(S_6\) with an amalgamated subgroup \(A_6\).
The strategy to prove the main theorem is to translate the two questions into the geometric language. Let \(\bar{G}\) be a finite subgroup, and let \(\tau: \bar{G}\rightarrow \mathrm{Cr}_3({\mathbb{C}})\) be a monomorphism. Then there is a birational map \(\xi: V \dashrightarrow {\mathbb{P}}^3\) such that
1. the threefold \(V\) is normal and has terminal singularities;
2. there exists a monomorphism \(v: \bar{G} \rightarrow \mathrm{Aut}(V)\);
3. \(\forall g\in \bar{G}\), we have \(\tau (g)=\xi\circ v(g)\circ \xi^{-1}\);
4. there is a \(v( \bar{G})-\)Mori fibration \(\pi: V\rightarrow S\), i.e., a non-birational \(v(\bar{G})-\) equivariant surjective morphism with connected fibers such that the divisor \(-K_V\) is \(\pi\)-ample, and for every \(v(\bar{G})-\)invariant Weil divisor \(D\) on \(V\), there is \(\delta\in {\mathbb{Q}}\) such that \[ \delta K_V+D\sim_{\mathbb{Q}} \pi^*(H) \] for some \({\mathbb{Q}}\)-Cartier divisor \(H\) on the variety \(S\).
The quadruple \((V, \xi, v, \pi)\) is called a Mori regularization of the pair \((\bar{G}, \tau)\).
If \(S\) is a point, then \(V\) is called \(v(\bar{G})\)-birationally rigid if for every Mori regularization \((V', \xi', v', \pi')\) of the the pair \((\bar{G}, \tau)\), we have \(V'\cong V\), \(\pi'(V')\) is a point, and the subgroups \(v(\bar{G})\) and \(v'(\bar{G})\) are conjugate in \(\mathrm{Aut}(V)\cong \mathrm{Aut}(V')\).
If \(S\) is a point, then \(V\) is called \(v(\bar{G})\)-birationally superrigid if for every Mori regularization \((V', \xi', v', \pi')\) of the pair \((\bar{G}, \tau)\), the map \(\xi^{-1}\circ \xi'\) is biregular.
Theorem 1.5 is an application of Theorem 1.24.
{ Theorem 1.24.} Suppose that \(\bar{G}=A_6\). If \(V\cong {\mathbb{P}}^3\), then the threefold \(V\) is \(\bar{G}\)-birationally rigid (but not \(\bar{G}\)-birationally superrigid) and \(\mathrm{Bir}^{\bar{G}} (V)\) is a free product of two copies of \(S_6\) with amalgamated subgroup \(A_6\). If \(V\) is either the Segre cubic or a smooth quadric threefold, then \(V\) is \(\bar{G}\)-birationally superrigid and \(\mathrm{Bir}^{\bar{G}} (V)\cong S_6\).
The authors introduce a new technique to prove Theorem 1.24. The main idea of the proof of Theorem 1.24 is to use the machinery of multiplier ideal (Kawamata subadjunction theorem, Nadel-Shokurov vanishing theorem, the Riemann-Roch theorem, the Clifford theorem and Castelnuovo bound).

MSC:

14J30 \(3\)-folds
14J70 Hypersurfaces and algebraic geometry
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
14B05 Singularities in algebraic geometry

Software:

Magma
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Allan Adler, On the automorphism group of a certain cubic threefold, Amer. J. Math. 100 (1978), no. 6, 1275 – 1280. · Zbl 0405.14019
[2] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. · Zbl 0559.14017
[3] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. · Zbl 0718.14023
[4] A.Beauville, Non-rationality of the symmetric sextic Fano threefold, arXiv:math/1102.1255 (2011). · Zbl 1317.14033
[5] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235 – 265. Computational algebra and number theory (London, 1993). · Zbl 0898.68039
[6] Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, London Mathematical Society Lecture Note Series, vol. 280, Cambridge University Press, Cambridge, 2000. · Zbl 0952.30001
[7] I. A. Chel\(^{\prime}\)tsov, Birationally rigid Fano varieties, Uspekhi Mat. Nauk 60 (2005), no. 5(365), 71 – 160 (Russian, with Russian summary); English transl., Russian Math. Surveys 60 (2005), no. 5, 875 – 965. · Zbl 1145.14032
[8] Ivan Cheltsov, Log canonical thresholds of del Pezzo surfaces, Geom. Funct. Anal. 18 (2008), no. 4, 1118 – 1144. · Zbl 1161.14030
[9] I. A. Chel\(^{\prime}\)tsov and K. A. Shramov, Log-canonical thresholds for nonsingular Fano threefolds, Uspekhi Mat. Nauk 63 (2008), no. 5(383), 73 – 180 (Russian, with Russian summary); English transl., Russian Math. Surveys 63 (2008), no. 5, 859 – 958. · Zbl 1167.14024
[10] I.Cheltsov, C.Shramov, On exceptional quotient singularities, Geometry and Topology, 15 (2011), 1843-1882 · Zbl 1232.14001
[11] C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281 – 356. · Zbl 0245.14010
[12] Alessio Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), no. 2, 223 – 254. · Zbl 0866.14007
[13] Alessio Corti, Singularities of linear systems and 3-fold birational geometry, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 259 – 312. · Zbl 0960.14017
[14] János Kollár, Karen E. Smith, and Alessio Corti, Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, vol. 92, Cambridge University Press, Cambridge, 2004. · Zbl 1060.14073
[15] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. · Zbl 0568.20001
[16] Igor V. Dolgachev and Vasily A. Iskovskikh, Finite subgroups of the plane Cremona group, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Inc., Boston, MA, 2009, pp. 443 – 548. · Zbl 1219.14015
[17] Walter Feit, The current situation in the theory of finite simple groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 55 – 93. · Zbl 0344.20008
[18] Hans Finkelnberg, Small resolutions of the Segre cubic, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), no. 3, 261 – 277. · Zbl 0628.14033
[19] Hans Finkelnberg and Jürgen Werner, Small resolutions of nodal cubic threefolds, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 2, 185 – 198. · Zbl 0703.14026
[20] Paul Hacking and Yuri Prokhorov, Smoothable del Pezzo surfaces with quotient singularities, Compos. Math. 146 (2010), no. 1, 169 – 192. · Zbl 1194.14054
[21] C.Hacon, J.McKernan, Ch.Xu, On the birational automorphisms of varieties of general type, arXiv:math/1011.1464 (2010)
[22] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[23] Bruce Hunt, The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, vol. 1637, Springer-Verlag, Berlin, 1996. · Zbl 0904.14025
[24] V. A. Iskovskih and Ju. I. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Mat. Sb. (N.S.) 86(128) (1971), 140 – 166 (Russian). · Zbl 0222.14009
[25] I. R. Shafarevich , Algebraic geometry. V, Encyclopaedia of Mathematical Sciences, vol. 47, Springer-Verlag, Berlin, 1999. Fano varieties; A translation of Algebraic geometry. 5 (Russian), Ross. Akad. Nauk, Vseross. Inst. Nauchn. i Tekhn. Inform., Moscow; Translation edited by A. N. Parshin and I. R. Shafarevich. V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1 – 247.
[26] V. A. Iskovskikh and A. V. Pukhlikov, Birational automorphisms of multidimensional algebraic manifolds, J. Math. Sci. 82 (1996), no. 4, 3528 – 3613. Algebraic geometry, 1. · Zbl 0917.14007
[27] Yujiro Kawamata, On Fujita’s freeness conjecture for 3-folds and 4-folds, Math. Ann. 308 (1997), no. 3, 491 – 505. · Zbl 0909.14001
[28] Yujiro Kawamata, Subadjunction of log canonical divisors. II, Amer. J. Math. 120 (1998), no. 5, 893 – 899. · Zbl 0919.14003
[29] János Kollár, Singularities of pairs, Algebraic geometry — Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221 – 287. · Zbl 0905.14002
[30] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals.
[31] Nicole Lemire, Vladimir L. Popov, and Zinovy Reichstein, Cayley groups, J. Amer. Math. Soc. 19 (2006), no. 4, 921 – 967. · Zbl 1103.14026
[32] Marco Manetti, Normal degenerations of the complex projective plane, J. Reine Angew. Math. 419 (1991), 89 – 118. · Zbl 0719.14023
[33] Massimiliano Mella, Birational geometry of quartic 3-folds. II. The importance of being \Bbb Q-factorial, Math. Ann. 330 (2004), no. 1, 107 – 126. · Zbl 1058.14022
[34] S. Mori and Yu. G. Prokhorov, Multiple fibers of del Pezzo fibrations, Tr. Mat. Inst. Steklova 264 (2009), no. Mnogomernaya Algebraicheskaya Geometriya, 137 – 151; English transl., Proc. Steklov Inst. Math. 264 (2009), no. 1, 131 – 145. · Zbl 1312.14048
[35] Shigeru Mukai, Curves and symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 1, 7 – 10. · Zbl 0768.14014
[36] Isidro Nieto, The normalizer of the level (2,2)-Heisenberg group, Manuscripta Math. 76 (1992), no. 3-4, 257 – 267. · Zbl 0778.20019
[37] K.Pettersen, On nodal determinantal quartic hypersurfaces in \( \mathbb{P}^4\), Thesis, University of Oslo (1998)
[38] Yu.Prokhorov, Simple finite subgroups of the Cremona group of rank \( 3\), Journal of Algebraic Geometry, to appear · Zbl 1257.14011
[39] Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345 – 414. · Zbl 0634.14003
[40] Jean-Pierre Serre, A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field, Mosc. Math. J. 9 (2009), no. 1, 193 – 208, back matter (English, with English and Russian summaries). · Zbl 1203.14017
[41] V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105 – 203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95 – 202. · Zbl 0785.14023
[42] C.Shramov, Birational automorphisms of nodal quartic threefolds, arXiv:math/0803.4348 (2008)
[43] Gang Tian, On Kähler-Einstein metrics on certain Kähler manifolds with \?\(_{1}\)(\?)>0, Invent. Math. 89 (1987), no. 2, 225 – 246. · Zbl 0599.53046
[44] Gang Tian and Shing-Tung Yau, Kähler-Einstein metrics on complex surfaces with \?\(_{1}\)>0, Comm. Math. Phys. 112 (1987), no. 1, 175 – 203. · Zbl 0631.53052
[45] J.A.Todd, Configurations defined by six lines in space of three dimensions, Mathematical Proceedings of the Cambridge Philosophical Society 29 (1933), 52-68 · Zbl 0006.12601
[46] Dana R. Vazzana, Invariants and projections of six lines in projective space, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2673 – 2688. · Zbl 0983.51002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.