### Abstract

Let X be a smooth, projective curve of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. I provide a conjectural formula for the degree of the scheme of dormant Projective General Linear PGL(r)-opers on X where r ≥ 2 (I assume that p is greater than an explicit constant depending on g, r). For r = 2, a dormant PGL(2)-oper is a dormant indigenous bundle on X in the sense of Shinichi Mochuzki (and his work provides a formula only for g = 2, r = 2, p ≥ 5, from a different point of view). In 2014, Yasuhiro Wakabayashi has shown that my conjectural formula holds for r = 2, g ≥ 2, and p > 2g - 2 and more recently he has proved the conjecture in all ranks for generic curves of genus at least two.

Original language | English (US) |
---|---|

Pages (from-to) | 2599-2613 |

Number of pages | 15 |

Journal | International Mathematics Research Notices |

Volume | 2017 |

Issue number | 9 |

DOIs | |

State | Published - May 1 2017 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Mathematics Research Notices*,

*2017*(9), 2599-2613. https://doi.org/10.1093/imrn/rnw066

**The degree of the dormant operatic locus.** / Joshi, Kirti N.

Research output: Contribution to journal › Article

*International Mathematics Research Notices*, vol. 2017, no. 9, pp. 2599-2613. https://doi.org/10.1093/imrn/rnw066

}

TY - JOUR

T1 - The degree of the dormant operatic locus

AU - Joshi, Kirti N

PY - 2017/5/1

Y1 - 2017/5/1

N2 - Let X be a smooth, projective curve of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. I provide a conjectural formula for the degree of the scheme of dormant Projective General Linear PGL(r)-opers on X where r ≥ 2 (I assume that p is greater than an explicit constant depending on g, r). For r = 2, a dormant PGL(2)-oper is a dormant indigenous bundle on X in the sense of Shinichi Mochuzki (and his work provides a formula only for g = 2, r = 2, p ≥ 5, from a different point of view). In 2014, Yasuhiro Wakabayashi has shown that my conjectural formula holds for r = 2, g ≥ 2, and p > 2g - 2 and more recently he has proved the conjecture in all ranks for generic curves of genus at least two.

AB - Let X be a smooth, projective curve of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. I provide a conjectural formula for the degree of the scheme of dormant Projective General Linear PGL(r)-opers on X where r ≥ 2 (I assume that p is greater than an explicit constant depending on g, r). For r = 2, a dormant PGL(2)-oper is a dormant indigenous bundle on X in the sense of Shinichi Mochuzki (and his work provides a formula only for g = 2, r = 2, p ≥ 5, from a different point of view). In 2014, Yasuhiro Wakabayashi has shown that my conjectural formula holds for r = 2, g ≥ 2, and p > 2g - 2 and more recently he has proved the conjecture in all ranks for generic curves of genus at least two.

UR - http://www.scopus.com/inward/record.url?scp=85021066954&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021066954&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnw066

DO - 10.1093/imrn/rnw066

M3 - Article

AN - SCOPUS:85021066954

VL - 2017

SP - 2599

EP - 2613

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 9

ER -