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

TL;DR

This paper explores the geometric properties of the Quot scheme parametrizing length l quotients of a sheaf on a smooth surface, focusing on singularities, intersection theory, and cohomology.

Contribution

It provides new insights into the structure and invariants of the Quot scheme on surfaces, including its singularities and intersection theory.

Findings

Analysis of the singularity structure of the Quot scheme

Results on intersection theory of the Quot scheme

Cohomological properties of sheaves on the Quot scheme

Abstract

We study the geometry of the Quot scheme $\mathrm{Quot}^l_{S}(\mathcal{E})$ of length $l$ coherent sheaf quotients of a locally free sheaf $\mathcal{E}$ on a smooth projective surface $\mathrm{S}$. In particular, we investigate the nature of its singularities, its intersection theory, and the cohomology of sheaves on $\mathrm{Quot}^l_{S}(\mathcal{E})$.