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

Samuel Stark

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. In particular, we investigate the nature of its singularities, its intersection theory, and the cohomology of sheaves on $\mathrm{Quot}^l_{S}(\mathcal{E})$.

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

Abstract

We study the geometry of the Quot scheme of length coherent sheaf quotients of a locally free sheaf on a smooth projective surface. In particular, we investigate the nature of its singularities, its intersection theory, and the cohomology of sheaves on .

Paper Structure

This paper contains 17 sections, 33 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Basic properties
  3. Quot schemes
  4. Tautological sheaves
  5. Obstruction theory
  6. Local properties
  7. Embedding dimension
  8. Conjectural description of the singularities
  9. Resolution of singularities
  10. Intersection theory
  11. Relations
  12. Integrals
  13. Higher rank Severi formula
  14. Cohomology
  15. Structure sheaf
  16. ...and 2 more sections

Key Result

Theorem 1

(i) There exists a morphism of schemes taking a length two subscheme $\mathrm{Z}$ of $\mathbf{P}(\mathcal{E})$ to $\mathcal{E}=p_{\ast} \mathcal{O}(1)\rightarrow p_{\ast} \mathcal{O}_{\mathrm{Z}}(1)$. (ii) The morphism $\rho^{2}$ is a resolution of singularities of $\operatorname{Quot}^2_{\mathrm{S}}(\mathcal{E})$. (iii) If $\mathcal{F}$

Theorems & Definitions (69)

  • Conjecture 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • ...and 59 more