On base change of the fundamental group scheme

TL;DR

This paper presents examples of smooth projective curves over fields of characteristic p where the base change of the fundamental group scheme fails, exploring the behavior of F-trivial vector bundles in families.

Contribution

It provides the first known examples demonstrating failure of base change for the fundamental group scheme in characteristic p, linking this to F-trivial vector bundles.

Findings

Examples of failure of base change for fundamental group schemes in characteristic p.

Analysis of F-trivial vector bundles in families.

Insights into the behavior of F-triviality in non-trivial families.

Abstract

We provide for all prime numbers $p$ examples of smooth projective curves over a field of characteristic $p$ for which base change of the fundamental group scheme fails. This is intimately related to how $F$-trivial vector bundles, i.e. bundles trivialized by a power of the Frobenius morphism, behave in (trivial) families. We conclude with a study of the behavior of $F$-triviality in (not necessarily trivial) families.