# Rank-one sheaves and stable pairs on surfaces

## TL;DR

This paper investigates the geometry of rank-one sheaves and stable pairs on surfaces, establishing a perfect obstruction theory and virtual fundamental class, with applications to enumerative geometry on the projective plane.

## Contribution

It introduces an embedding of the moduli space into a smooth space that yields a perfect obstruction theory and relates the virtual class to a vector bundle's Euler class, advancing the understanding of stable pairs on surfaces.

## Key findings

- The moduli space admits an embedding into a smooth ambient space.
- The virtual fundamental class equals the Euler class of a vector bundle.
- On , the expected count matches the actual length of the Quot scheme.

## Abstract

We study rank-one sheaves and stable pairs on a smooth projective complex surface. We obtain an embedding of the moduli space of limit stable pairs into a smooth space. The embedding induces a perfect obstruction theory, which, over a surface with irregularity 0, agrees with the usual deformation-obstruction theory. The perfect obstruction theory defines a virtual fundamental class on the moduli space. Using the embedding, we show that the virtual class equals the Euler class of a vector bundle on the smooth ambient space. As an application, we show that on $\mathbb{P}^2$, the expected count of the finite Quot scheme in arXiv:1610.04185 is its actual length. We also obtain a universality result for tautological integrals on the moduli space of stable pairs.