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Opers of higher types, Quot-schemes and Frobenius instability loci

Kirti Joshi, Christian Pauly

Abstract

In this paper we continue our study of the Frobenius instability locus in the coarse moduli space of semi-stable vector bundles of rank $r$ and degree $0$ over a smooth projective curve defined over an algebraically closed field of characteristic $p>0$. In a previous paper we identified the "maximal" Frobenius instability strata with opers (more precisely as opers of type $1$ in the terminology of the present paper) and related them to certain Quot-schemes of Frobenius direct images of line bundles. The main aim of this paper is to describe for any integer $q \geq 1$ a conjectural generalization of this correspondence between opers of type $q$ (which we introduce here) and Quot-schemes of Frobenius direct images of vector bundles of rank $q$. We also give a conjectural formula for the dimension of the Frobenius instability locus.

Opers of higher types, Quot-schemes and Frobenius instability loci

Abstract

In this paper we continue our study of the Frobenius instability locus in the coarse moduli space of semi-stable vector bundles of rank and degree over a smooth projective curve defined over an algebraically closed field of characteristic . In a previous paper we identified the "maximal" Frobenius instability strata with opers (more precisely as opers of type in the terminology of the present paper) and related them to certain Quot-schemes of Frobenius direct images of line bundles. The main aim of this paper is to describe for any integer a conjectural generalization of this correspondence between opers of type (which we introduce here) and Quot-schemes of Frobenius direct images of vector bundles of rank . We also give a conjectural formula for the dimension of the Frobenius instability locus.

Paper Structure

This paper contains 8 sections, 19 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Quot-schemes
  3. A finiteness theorem
  4. Opers of type $q$
  5. Properties of the Quot-scheme ${{\rm Quot}^{r,0}(\mathit{F}_*({Q}))}$ when $q=1$
  6. The Quot-scheme ${{\rm Quot}^{r,0}(\mathit{F}_*({Q}))}$ is bigger than expected when $q \geq 2$
  7. A characterization of dormant opers of type $q$
  8. Dimension of Frobenius instability loci

Key Result

Proposition 2.4

Let $r\geq 1$ be an integer. Let $Q$ be a vector bundle on $X$ of rank $q\leq r-1$ and slope $\mu(Q)=\mu$. Let $[E]\in {{\rm Quot}^{r,0}(\mathit{F}_*({Q}))}$ be a point of the Quot-scheme. Then the following assertions hold.

Theorems & Definitions (50)

  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.5
  • proof
  • Remark 3.6
  • Remark 3.7
  • ...and 40 more