Matrix factorization for quasi-homogeneous singularities

TL;DR

This paper explores the structure of reflexive sheaves on quasi-homogeneous singularities, linking them to divisors, and classifies matrix factorizations, confirming a conjecture on point modules.

Contribution

It establishes a group isomorphism between reflexive sheaves and divisors, and reduces matrix factorization problems to rank one cases, proving a conjecture.

Findings

Established a group isomorphism for reflexive sheaves and divisors.

Reduced matrix factorization problem to rank one reflexive sheaves.

Enumerated all matrix factorizations for rank one reflexive sheaves.

Abstract

Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by $\mathbb{C}^*$-divisors, modulo linear equivalence. When $\dim(X)=2$ we reduce the problem of finding matrix factorizations of arbitrary reflexive $\mathcal{O}_X$-modules to the same question on rank one reflexive sheaves. We then enumerate the matrix factorizations of all rank one reflexive sheaves. As a consequence, we prove a conjecture of Etingof and Ginzburg on point modules.