### Abstract

Let X be a smooth projective curve of genus g ≥2 defined over an algebraically closed field k of characteristic p>0. For p>r(r-1)(r-2)(g-1) we construct an atlas for the locus of all Frobenius-destabilized bundles of rank r (i.e. we construct all Frobenius-destabilized bundles of rank r and degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F^{*}(E) under the Frobenius morphism of X has maximal Harder-Narasimhan polygon and the set of opers having zero p-curvature. We also show that, after fixing the determinant, these sets are finite, which enables us to derive the dimension of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X. The finiteness is proved by studying the properties of the Hitchin-Mochizuki morphism; an alternative approach to finiteness has been realized in [3]. In particular we prove a generalization of a result of Mochizuki to higher ranks.

Original language | English (US) |
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Pages (from-to) | 39-75 |

Number of pages | 37 |

Journal | Advances in Mathematics |

Volume | 274 |

DOIs | |

State | Published - Apr 9 2015 |

### Keywords

- Frobenius map
- Local system
- Moduli spaces
- Primary
- Secondary
- Vector bundles

### ASJC Scopus subject areas

- Mathematics(all)