The variation of the monodromy group in families of stratified bundles in positive characteristic

TL;DR

This paper investigates how the monodromy group of stratified bundles varies in smooth families over fields of positive characteristic, proving the strong form of a conjecture over uncountable fields and exploring generalizations.

Contribution

It strengthens the positive equicharacteristic p-curvature conjecture by proving the strong form for uncountable fields and analyzing possible extensions for countable fields.

Findings

Strong form of the p-curvature conjecture holds over uncountable fields.

Counterexamples or limitations are identified for countable fields.

Provides insights into the behavior of monodromy groups in positive characteristic.

Abstract

In this article we study smooth families of stratified bundles in positive characteristic and the variation of their monodromy group.Our aim is, in particular, to strengthen the weak form of the positive equicharacteristic $p$-curvature conjecture stated and proved by Esnault and Langer in "On a positive equicharacteristic variant of the $p$-curvature conjecture" (Doc. Math. 18 (2013)). The main result is that if the ground field is uncountable then the strong form holds. In the case where the ground field is countable we provide positive and negative answers to possible generalizations.