Counting stable sheaves on singular curves and surfaces

TL;DR

This paper develops a method to analyze virtual cycles on moduli spaces of stable sheaves over singular curves and surfaces, relating them through affine bundles and refined Gysin homomorphisms, with applications to algebraic geometry.

Contribution

It introduces a technique to construct affine bundles with perfect obstruction theories relative to a base, enabling the study of virtual cycles on moduli spaces of sheaves on singular varieties.

Findings

Constructed affine bundles with relative perfect obstruction theories.

Linked virtual cycles via refined Gysin homomorphisms.

Applied framework to moduli spaces on surfaces and Fano threefolds.

Abstract

Given a quasi-projective scheme M over complex numbers equipped with a perfect obstruction theory and a morphism to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M'/ M that admits a perfect obstruction theory relative to B. We study the resulting virtual cycles on the fibers of M'/B and relate them to the image of the virtual cycle [M]^vir under refined Gysin homomorphisms. Our main application is when M is a moduli space of stable codimension 1 sheaves with a fixed determinant L on a nonsingular projective surface or Fano threefold, B is the linear system |L|, and the morphism to B is given by taking the divisor associated to a coherent sheaf.