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Infinitesimal deformations of some Quot schemes

Indranil Biswas, Chandranandan Gangopadhyay, Ronnie Sebastian

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 $\mathcal{Q}(E,d)$.

Infinitesimal deformations of some Quot schemes

Abstract

Let be a vector bundle on a smooth complex projective curve of genus at least two. Let be the Quot scheme parameterizing the torsion quotients of of degree . We compute the cohomologies of the tangent bundle . In particular, the space of infinitesimal deformations of is computed. Kempf and Fantechi computed the space of infinitesimal deformations of . We also explicitly describe the infinitesimal deformations of .
Paper Structure (13 sections, 35 theorems, 210 equations)

This paper contains 13 sections, 35 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Some tangent bundle sequences
  3. Atiyah sequence
  4. The relative Atiyah sequence
  5. Infinitesimal deformation map
  6. Canonical bundle of $S_d$
  7. Projective bundle computations
  8. Cohomology of some sheaves on $S_d$
  9. On the geometry of $\mathcal{Q}$
  10. Notation
  11. Direct image of sheaves on $C\times \mathcal{Q}$
  12. The tangent bundle of ${\mathcal{Q}}$
  13. Computation of cohomologies of $T_{\mathcal{Q}}$

Key Result

Theorem 1.1

Let $r\,=\, {\rm rank}(E)\,\geqslant \,2$. Then

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • ...and 57 more