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Deformations of free and linear free divisors. (Déformations de diviseurs libres et linéaires libres.) (English. French summary) Zbl 1301.14004
A reduced divisor \(D=V(f)\subseteq \mathbb C^n\) is called free if the sheaf \(\text{Der}(-\log D)\) of logarithmic vector fields is a locally free \(\mathcal{O}_{\mathbb C^n}\)-module. It is called linear if, furthermore, \(\text{Der}(-\log D)\) is globally generated by a basis consisting of vector fields whose all of coefficients, with respect to the standard basis \(\partial /\partial x_1, \cdots, \partial/\partial x_n\) of the space \(\text{Der}_{\mathbb C^n}\), are linear functions. Free divisors were introduced by K. Saito [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265–291 (1980; Zbl 0496.32007)] and linear free divisors were introduced by R.-O. Buchweitz and D. Mond [Lond. Math. Soc. Lect. Note Ser. 324, 41–77 (2006; Zbl 1101.14013)].
In the paper under review, the author describes the spaces of infinitesimal deformations and obstructions of a germ of a (linear) free divisor and performed calculations for some concrete examples. The main results of this paper can be summarized as follows:
Theorem A. (see Corollary 3.45). Let \((D,0) \subseteq (\mathbb C^n,0)\) be a germ of a reductive linear free divisor \(D\). Then it is formally rigid.
This is equivalent to say that for a germ of a reductive linear free divisor, there are no non-trivial families, at least on the level of formal power series.
Theorem B. (see Corollary 3.50) Let \((D,0) \subseteq (\mathbb C^n,0)\) be a germ of a free divisor generated by a weighted homogeneous polynomial. Then the functor \(\mathbf{FD}_D\) has a hull, i.e. \((D,0) \) has a formally versal deformation.
Here the functor \(\mathbf{FD}_D\) associates each local Artin ring \(A\) to the set of isomorphism classes of admissible deformations of \((D,0)\) over \(\text{Spec} A\).
Theorem C. (see Corollary 4.24) Let \((D,0) \subseteq (\mathbb C^n,0)\) be a germ of a Koszul free divisor. If we can put a logarithmic connection on \(\text{Der}_{\mathbb C^n}\) and \(\text{Der}(-\log D)\), then \(\mathbf{FD}_D\) has a hull.

MSC:
14B07 Deformations of singularities
13D10 Deformations and infinitesimal methods in commutative ring theory
14F40 de Rham cohomology and algebraic geometry
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