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The Degree of the Dormant Operatic Locus

2016; Oxford University Press; Linguagem: Inglês

10.1093/imrn/rnw066

1687-0247

Kirti Joshi,

North African History and Literature

Hara-naka ya Mono nimo tsukazu Naku hibari Matsuo Basho[14] Let |$X$| be a smooth, projective curve of genus |$g\geq 2$| 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 |${\rm PGL}(r)$|-opers on |$X$| where |$r\geq 2$| (I assume that |$p$| is greater than an explicit constant depending on |$g,r$|⁠). For |$r=2$|⁠, a dormant |${\rm PGL}(2)$|-oper is a dormant indigenous bundle on |$X$| in the sense of Shinichi Mochuzki (and his work provides a formula only for |$g=2,r=2,p\geq 5$|⁠, from a different point of view). In 2014, Yasuhiro Wakabayashi has shown that my conjectural formula holds for |$r=2,g\geq 2$|⁠, and |$p>2g-2$| and more recently he has proved the conjecture in all ranks for generic curves of genus at least two.