A $\mathrm{ PGL}_n^{ (N)}$-oper is a specific type of flat $\mathrm{ PGL}_n$-bundle on an algebraic curve in prime characteristic $p$ enhanced by an action of the sheaf of differential operators of level $N-1$. In this paper, we introduce and study a higher-level generalization of the Hitchin-Mochizuki morphism on the moduli space of $\mathrm{ PGL}_n^{ (N)}$-opers, defined via the characteristic polynomials of their $p^N$-curvatures. As an application, we prove the irreducibility of the moduli space classifying pointed stable curves equipped with dormant $\mathrm{ PGL}_2^{ (N)}$-opers, i.e., $\mathrm{ PGL}_2^{ (N)}$-opers with vanishing $p^N$-curvature.
Categories: math.AG