### 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) |
---|---|

Pages (from-to) | 39-75 |

Number of pages | 37 |

Journal | Advances in Mathematics |

Volume | 274 |

DOIs | |

State | Published - Apr 9 2015 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Hitchin-Mochizuki morphism, opers and Frobenius-destabilized vector bundles over curves.** / Joshi, Kirti N; Pauly, Christian.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 274, pp. 39-75. https://doi.org/10.1016/j.aim.2015.01.004

}

TY - JOUR

T1 - Hitchin-Mochizuki morphism, opers and Frobenius-destabilized vector bundles over curves

AU - Joshi, Kirti N

AU - Pauly, Christian

PY - 2015/4/9

Y1 - 2015/4/9

N2 - 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.

AB - 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.

KW - Frobenius map

KW - Local system

KW - Moduli spaces

KW - Primary

KW - Secondary

KW - Vector bundles

UR - http://www.scopus.com/inward/record.url?scp=84921947021&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84921947021&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2015.01.004

DO - 10.1016/j.aim.2015.01.004

M3 - Article

VL - 274

SP - 39

EP - 75

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -