A module-theoretic characterization of algebraic hypersurfaces. (English) Zbl 1403.14079

Miranda-Neto, Cleto B.

Can. Math. Bull. 61, No. 1, 166-173 (2018).

In the introduction, “some of the numerous references” for the past 4 decades came under criticism of the author because “most of the techniques have focused mainly on detecting families of examples and their features...” By contrast, the author’s “goal is to report the following surprising result, which had gone unnoticed and in fact turns out to be a new module-theoretic characterization of hypersurfaces”. More precisely, his main result is formulated as follows: the module of tangent vector fields to an affine algebraic variety over the field of characteristic zero is reflexive if and only if the variety is a hypersurface.
Reviewer’s remark: It should be noted that this fact is a very particular case of the well-known statement concerning the behavior of reflexive sheaves on normal varieties (see, e.g., Corollary 1.5 in [R. Hartshorne, Math. Ann. 254, 121–176 (1980; Zbl 0431.14004)]). More precisely, if \(\mathcal G\) is a subsheaf of a reflexive sheaf \(\mathcal F\) given on a normal variety, then \(\mathcal G\) is reflexive if and only if the set of associated primes of the quotient \(\mathcal F/\mathcal G\) consists of points of codimensions 0 and 1 only.

Reviewer: Aleksandr G. Aleksandrov (Moskva)

MSC:

14J70	Hypersurfaces and algebraic geometry
13N15	Derivations and commutative rings
32S22	Relations with arrangements of hyperplanes
13C05	Structure, classification theorems for modules and ideals in commutative rings
13C10	Projective and free modules and ideals in commutative rings
14N20	Configurations and arrangements of linear subspaces
14C20	Divisors, linear systems, invertible sheaves
32M25	Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions

Keywords:

reflexive modules; hypersurfaces; tangent vector fields; logarithmic vector fields; free divisors