Singularities of free divisors: algebra and geometry

Singularity theory is a branch of algebraic geometry concerned with the understanding of the geometry of zero-sets of algebraic equations near singular points, that is, points where the set is not a manifold. Free divisors are hypersurfaces in complex manifolds, which are "maximally" singular, in the sense that they are non-normal and their singular locus is Cohen-Macaulay. The goal of this project is to investigate normalization, (non-commutative) desingularization and singularity invariants of algebraic (analytic) varieties, where the focus lies on free divisors in complex manifolds. Moreover positive characteristic methods will be studied. The study of singularities of free divisors is fundamental for the understanding of singularities of more general analytic varieties. The main objectives of this work are: analyze and improve the process of normalization of free (and more general) divisors (improve the Grauert- Remmert normalization algorithm, as implemented e.g. in SINGULAR), find a natural way to desingularize free divisors and a measure for the distance of an arbitrary free divisor to the normal crossing divisor (study of blowups of free divisors, canonical desingularization in the case of discriminants), study the question of existence of a non-commutative desingularization of a free divisor, investigation of singularity invariants, in particular to characterize the normal crossing divisor (Torsion differentials, Bernstein-Sato polynomials), positive characteristic methods (definition of free divisors over fields of prime characteristic, singularity invariants), as a byproduct, visualize the resolution of singularities and normalization of free surfaces.

Algebraic varieties are geometric objects which are a main object of study of the field of algebraic geometry. A singularity of an algebraic variety (=a zeroset of polynomial equations) is a point where the variety is not smooth", i.e., has self-intersections or cusps. A free divisor is a hypersurface in a complex manifold, which is \maximally" singular, in the sense that it is non-normal and the singular locus has the structure of a maximal Cohen{Macaulay module.The main goal of this Erwin-Schrödinger fellowship was to study and understand singularities of free divisors and more general non-normal algebraic varieties. In particular the project was concerned with (non-commutative) desingularizations and characterizations of singularities of free divisors.Non-commutative resolutions (NCRs) of singularities are a very recent active area of research, in particular one is interested in the studying and constructing NCRs. They are also appearing in different contexts, for example, in representation theory, mathematical physics, (non-) commutative algebra and algebraic geometry.In this project, non-commutative desingularizations for non-normal rings were defined and also a new homological invariant was introduced and studied, the global spectrum of a singularity.This invariant is a set of natural numbers associated to the singularity, i.e., it is the set of all possible finite global dimensions of NCRs of a given singularity. We showed that simple surface singularities are characterized by their global spectra and determined global spectra of certain curves and surfaces with representation theoretic methods. The global spectrum is expected to have connections to other birational invariants, e.g., the Orlov spectrum, and will be object of further studies.Another line of research was the study the McKay correspondence for finite reflection groups. Here we were able to show a similar result as Auslander's version of the McKay correspondence involving the skew group ring. In particular we obtained NCRs of the discriminants of the reflection groups. These discriminants are free divisors.In the research goal characterization of singularities we studied a generalization of transversal intersection for singular varieties, so-called splayed intersections. We could show that certain conjectured formulas for characteristic classes holds for this class of singularities.Moreover, together with J. Bell, R.-O. Buchweitz and C. Ingalls, we organized a workshop at the Fields Institute in Toronto about topics with which this project was concerned: connections between commutative algebra, noncommutative geometry and representation theory.