The Degree of the Dormant Operatic Locus

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 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. Hara-naka ya Mono nimo tsukazu Naku hibariMatsuo Basho [Miura, 1991]

Frobenius-Ehresmann structures and Cartan geometries in positive characteristic

“…The former assertion is a direct consequence of Lemma 3.11.3 described below. The latter assertion follows from the former assertion, (26), and Proposition 3.10.5.…”

“…2 )-structures on a general smooth proper curve; this is a generalization of Joshi's conjecture (cf. [26,Conjecture 8.1]) for the rank 2 case, which was proved in [51].…”

“…Let us consider assertion (i). The problem is reduced to the case of N ∈ Z >0 because the assertion for N = ∞ follows immediately from this case together with (26) and (190). First, we shall construct a map of sets F N -Ehr (G,G),X/S → B (N ),s,t X/S .…”

“…By taking account of (200) and its proof, we expect that the following conjectural formula will hold for any large N; this conjecture may be thought of as a generalization of Joshi's conjecture for the rank 2 case (cf. [26,Conjecture 8.1]).…”

On Maximally Frobenius Destabilised Vector bundles

“…If r = p, then F X * (L ) is a maximally Frobenius destabilised rank-p stable vector bundle for any line bundle L on an arbitrary smooth projective curve X of genus g ≥ 2 in characteristic p > 0 (see [5] and [11]). If r < p, Zhao [11,Proposition 2.14] showed that for any given natural numbers p > 0, g ≥ 2 and r > 0 with r < p and p g − 1, there exists some maximally Frobenius destabilised rank-r stable vector bundle over some smooth projective curve of genus g ≥ 2 in characteristic p. Under the assumption p > r(r − 1)(r − 2)(g − 1), Joshi and Pauly [3] gave a correspondence between maximally Frobenius destabilised L. Li [2] stable vector bundles of degree 0 and dormant operatic loci, and proved the existence of Frobenius destabilised stable vector bundles of rank r and degree 0. Further results about Frobenius destabilised stable vector bundles can be found in [4,6,7] and [8].…”