Cosection localization and the Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$

TL;DR

This paper investigates the virtual intersection theory of Quot schemes of sheaf quotients on surfaces, using cosection localization to relate their virtual classes to those on canonical curves, and establishes formulas for virtual Euler characteristics.

Contribution

It introduces a cosection localization approach to analyze the virtual fundamental class of Quot schemes on surfaces, relating it to subschemes on canonical curves and deriving new formulas for virtual invariants.

Findings

Virtual fundamental class of Quot scheme equals that of a subscheme on a canonical curve up to sign.

Established a structure theorem for virtual tautological integrals on Quot schemes.

Proved the equality of virtual Euler characteristics for Quot schemes of different sheaves.

Abstract

Let $\mathcal{E}$ be a locally free sheaf of rank $r$ on a smooth projective surface $S$. The Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ of length $l$ coherent sheaf quotients of $\mathcal{E}$ is a natural higher rank generalization of the Hilbert scheme of $l$ points of $S$. We study the virtual intersection theory of this scheme. If $C\subset S$ is a smooth canonical curve, we use cosection localization to show that the virtual fundamental class of $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ is $(-1)^{l}$ times the fundamental class of the smooth subscheme $\mathrm{Quot}^{l}_{C}(\mathcal{E}\vert_{C})\subset\mathrm{Quot}^{l}_{S}(\mathcal{E})$. We then prove a structure theorem for virtual tautological integrals over $\mathrm{Quot}^{l}_{S}(\mathcal{E})$. From this we deduce, among other things, the equality of virtual Euler characteristics $\chi^{\mathrm{vir}}(\mathrm{Quot}^{l}_{S}(\mathcal{E}))=\chi^{\mathrm{vir}}(\mathrm{Quot}^{l}_{S}(\mathcal{O}^{\oplus r}))$.