Rank-one sheaves and stable pairs on surfaces

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.

29 pages. In this new version, we extend one of our main theorems to more cases. Hence, we have changed the title. We also fix a mistake in Proposition 7.2 and one in the proof of Proposition 4.1 in the first version. Comments are welcome!