On Cohen-Macaulay modules over non-commutative surface singularities

We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame.     

Brieskorn modules and Gauss Manin systems for non-isolated hypersurface singularities

We study the Brieskorn modules associated to a germ of a holomorphicfunction with non-isolated singularities and show that the Brieskornmodule has naturally the structure of a module over the ringof microdifferential operators of non-positive degree, and thatthe kernel of the morphism to the Gauss–Manin system coincideswith the torsion part for the action of t and also with thatfor the action of the inverse of the Gauss–Manin connection.This torsion part is not finitely generated in general, anda sufficient condition for the finiteness is given here. A Thom–Sebastiani-typetheorem for the sheaf of Brieskorn modules is also proved whenone of two functions has an isolated singularity.     

Gluing Cohen-Macaulay modules with applications to quasihomogeneous complete intersections with isolated singularities

Supported in part by DFG     

Generic hypersurface singularities

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x₁,...,x_r be closed points in general position in projective spacePn, then the linear subspaceV ofH⁰ (?ⁿ,O(d)) (the space of homogeneous polynomials of degreed on ?ⁿ) formed by those polynomials which are singular at eachx_i, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x₁,...,x_r. As such, the “expected” value for the dimension ofV is max(0,h⁰(O(d))?r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.     

On nonisolated hypersurface singularities

We study germs of holomorphic functions whose singular sets are hypersurfaces with isolated singularity in the cases where the transversal singularity is A ₁. For these singularities, we completely describe the homotopy structure of the Milnor fibers. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

Free deformations of hypersurface singularities

The article is devoted to the study of the classification problem for Saito free divisors making use of the deformation theory of varieties. In particular, in the quasihomogeneous case, we describe an approach for computation of free deformations of quasicones over quasismooth varieties based on properties of deformations of varieties with $ {\mathbb{G}_m} $ -action. We also discuss some applications including the problem of compactification of modular spaces and computation of free deformations for certain simple, unimodal, and unimodular singularities.     

Equivalences between isolated hypersurface singularities

Research by this author supported in part by N.S.F. Grant No. DMS-84114477     

Bezout's theorem and Cohen-Macaulay modules

We define very proper intersections of modules and projective subschemes. It turns out that equidimensional locally Cohen-Macaulay modules intersect very properly if and only if they intersect properly. We prove a Bezout theorem for modules which meet very properly. Furthermore, we show for equidimensional subschemes X and Y: If they intersect properly in an arithmetically Cohen-Macaulay subscheme of positive dimension then X and Y are arithmetically Cohen-Macaulay. The module version of this result implies splitting criteria for reflexive sheaves. Received August 26, 1999 / Published online March 12, 2001     

Maximal multihomogeneity of algebraic hypersurface singularities

From the degree zero part of the logarithmic vector fields along analgebraic hypersurface singularity we identify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one–to–one correspondence to maximal tori in the linear jet of the embedded automorphism group. These results are motivated by Kyoji Saito’s characterization of quasihomogeneity for isolated hypersurface singularities [Saito in Invent. Math. 14, 123–142 (1971)] and extend previous work with Granger and Schulze [Compos. Math. 142(3), 765–778 (2006), Theorem 5.4] and of Hauser and Müller [Nagoya Math. J. 113, 181–186 (1989), Theorem 4].