Infinitesimal deformations of some Quot schemes

TL;DR

This paper investigates the infinitesimal deformations of Quot schemes parameterizing torsion quotients of vector bundles on complex curves, extending previous work on symmetric products by explicitly computing their deformation spaces.

Contribution

It computes the cohomologies of the tangent bundle of Quot schemes for vector bundles on curves and explicitly describes their infinitesimal deformations, generalizing known results.

Findings

Cohomology of tangent bundle of Quot schemes computed

Infinitesimal deformation space explicitly described

Extension of deformation results from symmetric products to general Quot schemes

Abstract

Let $E$ be a vector bundle on a smooth complex projective curve $C$ of genus at least two. Let $\mathcal{Q}(E,d)$ be the Quot scheme parameterizing the torsion quotients of $E$ of degree $d$. We compute the cohomologies of the tangent bundle $T_{\mathcal{Q}(E,d)}$. In particular, the space of infinitesimal deformations of $\mathcal{Q}(E,d)$ is computed. Kempf and Fantechi computed the space of infinitesimal deformations of $\mathcal{Q}(\mathcal{O}_C,d)\,=\, C^{(d)}$. We also explicitly describe the infinitesimal deformations of $\mathcal{Q}(E,d)$.