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Plane algebroid curves in arbitrary characteristic. (English) Zbl 1508.14002

IMPAN Lecture Notes 4. Warsaw: Polish Academy of Science, Institute of Mathematics (ISBN 978-83-86806-51-5/pbk). 134 p. (2022).
The authors chronicle the main results on plane algebroid curves over fields \(k\) of arbitrary characteristic \(p \geq 0\). The work described covers many decades, so this book will be valuable to the reader wanting to learn about the subject without having to collect the many papers and books it is based on. Below we summarize the material of the six chapters.
Chapter 1: Weierstrass theorem and applications. The authors give background results on power series rings \(R = k[[x_1, \dots, x_n]]\) in \(n\) variables, based on [N. Bourbaki, Éléments de mathématique. Paris: Hermann & Cie (1962; Zbl 0142.00102); O. Zariski and P. Samuel, Commutative algebra. Vol. II. Princeton, N.J.-Toronto-London-New York: D. Van Nostrand Company, Inc. (1960; Zbl 0121.27801); H. Grauert and R. Remmert, Analytische Stellenalgebren. Unter Mitarbeit von O. Riemenschneider. (Analytic place algebras). Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0231.32001); T. de Jong and G. Pfister, Local analytic geometry. Basic theory and applications. Braunschweig: Vieweg (2000; Zbl 0959.32011)]. The topics include results on summable families of power series, the Weierstrass division and preparation theorems, factorization results (including the fact that \(R\) is a UFD), the implicit and inverse function theorems, and Hensel’s lemma. The last section gives some specialized results in the case of \(n=2\) variables.
Chapter 2: Plane algebraic curves (basic concepts). The main topic of the book begins here, plane algebroid curves given by a power series \(f(x,y) \in k[[x,y]]\) with \(f(0,0)=0\). If \(f(x,y) = f_1^{m_1} \dots f_r^{m_r}\) with \(f_i\) irreducible and pairwise coprime, then the curves \(\{f_i = 0\}\) are the irreducible components with multiplicities \(m_i\). One result is the local normalization theorem giving a resolution of singularity by a finite sequence of quadratic transformations due to M. Noether [Rend. Circ. Mat. Palermo 4, 89–108 (1890; JFM 22.0712.02)] and O. Zariski [Am. J. Math. 87, 507–536 (1965; Zbl 0132.41601)]. Intersection multiplicities are defined in terms of valuations \(v_f\) associated to branches \(f\) and the semigroup associated to a branch \(f\) is the set \(\Gamma (f) \subset \mathbb N\) of all \(v_f (h)\) taken over all power series \(h\) with \(h \not \in (f)\). Two branches \(f,g\) are equisingularity if \(\Gamma (f) = \Gamma (g)\), the same as (a)-equivalence in the sense of Zariski ([Acevedo, , Fort Wayne, IN: Purdue University (PhD Thesis) (1967)] and R. Waldi [Commun. Algebra 28, No. 9, 4389–4401 (2000; Zbl 0969.14019)]. Log-distance and Newton diagrams are introduced.
Chapter 3: Plane algebroid branches. After giving generalities on (virtual) conductors and minimal systems of generators for numerical semigroups, the authors study the structure of the semigroup \(\Gamma (f)\) associated to a branch, following A. Seidenberg [Trans. Am. Math. Soc. 57, 387–425 (1945; Zbl 0060.07101)]. Existence of the key polynomials is proved and the Zariski characteristic sequence of a branch is introduced. Tools used are the strong triangle inequality and log distance between branches developed by E. García Barroso and A. Płoski [Rev. Mat. Complut. 28, No. 1, 227–252 (2015; Zbl 1308.32032); Colloq. Math. 156, No. 2, 243–254 (2019; Zbl 1434.32042)]. Following O. Zariski [in: C.I.M.E. 3. Ciclo Varenna 1969, Quest. algebr. Varieties, 261–343 (1970; Zbl 0204.54503)], the key polynomials are also constructed from the Puiseux sequence. The chapter closes with the characterization of semigroups associated with branches due to H. Bresinsky [Proc. Am. Math. Soc. 32, 381–384 (1972; Zbl 0218.14003)] and G. Angermüller [Math. Z. 153, 267–282 (1977; Zbl 0331.14015)] and the intersection formula.
Chapter 4: Equisingularity invariants. Following the book of A. Campillo [Algebroid curves in positive characteristic. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0451.14010)], the authors study the classical invariants of plane curve singularities: the degree of the conductor \(c\), the delta invariant \(\delta\), and the Milnor number \(\mu\) (in arbitrary characteristic it is defined by \(\mu = c-r+1\), where \(r\) is the number of irreducible components). They relate the semigroup of a plane branch to the multiplicity sequence and prove that two branches are equisingular if and only if they have the same multiplicity sequence. The chapter closes with the Hamburger-Noether expansions as a method to study algebroid curves in arbitrary characteristic, which were introduced by G. Ancochea [Courbes algébriques sur corps fermés de caractéristique quelconque. Acta Salamantic., Ci., Sec. Mat. 1 (1946; Zbl 0063.00081)] to avoid using the characteristic zero Puiseux series.
Chapter 5: Simple curve singularities. In characteristic zero, V. I. Arnol’d classified simple singularities as the famous A-D-E-singularities [Russ. Math. Surv. 29, No. 2, 10–50 (1974; Zbl 0304.57018)]. G. M. Greuel and H. Kröning [Math. Z. 203, No. 2, 339–354 (1990; Zbl 0715.14001)] modified the characterization of simple singularities to positive characteristic, using the definition of K. Kiyek and G. Steinke [Arch. Math. 45, 565–573 (1985; Zbl 0553.14012)]. Here the authors give a new proof using the theorem of finite determinacy and the consequence that an isolated plane curve singularity given by \(f=0\) is determined by its \(2 \tau\)-jet in the sense of contact equivalence, where \(\tau\) is the Tjurina number of \(f\). The answer depends on whether char \(k \geq 7\) or char \(k \leq 5\).
Chapter 6: Computational aspects. Using standard bases and Sagbi bases, the authors show how computer algebra systems can be used to compute singularity invariants. They illustrate with examples using SINGULAR.

MSC:

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14H50 Plane and space curves
14H25 Arithmetic ground fields for curves

Software:

SINGULAR