For every level N, the moduli space of pointed stable curves with a dormant PGL_2^{(N)}-oper is irreducible when nonempty, and the space of p^N-nilpotent opers is connected.
Projective and affine structures in positive characteristic I: Chern class formulas and Characterizations of projective spaces
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abstract
This paper aims to develop a theory of projective and affine structures on higher-dimensional varieties in positive characteristic. This theory deals with Frobenius-projective and Frobenius-affine structures, which have been previously investigated in the case where the underlying space is a curve. We first provide a description of such structures in terms of Berthelot's higher-level differential operators. That description leads us to obtain a positive characteristic version of Gunning's formulas, which give necessary conditions on Chern classes for the existence of Frobenius-projective and Frobenius-affine structures, respectively. Finally, we establish some characterizations of projective spaces using Frobenius-projective structures.
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The irreducibility of the moduli space of pointed stable curves with dormant $\mathrm{PGL}_2^{(N)}$-oper
For every level N, the moduli space of pointed stable curves with a dormant PGL_2^{(N)}-oper is irreducible when nonempty, and the space of p^N-nilpotent opers is connected.