Operators with nilpotent $p$-curvature

Abstract:

This article studies the variety of nilpotent $p$-curvature of linear differential operators on the sphere with $m + 1$ singularities. We shall show that for a given set of exponents $\in {{\mathbf {F}}_p}$ satisfying Fuchs’s relation the associated variety of nilpotent $p$-curvature is a nonempty complete intersection of dimension, essentially, $m - 2$. The subject has previously been studied by Katz, Honda, and Dwork.