A $p$-adic Bertini theorem for unipotent local systems

TL;DR

This paper proves a Bertini-type theorem for unipotent rigid fundamental groups, showing that for certain varieties over fields of characteristic p, there exist curves inducing surjective maps on fundamental groups.

Contribution

It introduces a version of Bertini's theorem specifically for unipotent rigid fundamental groups in positive characteristic.

Findings

Existence of curves with surjective fundamental group maps

Extension of Bertini's theorem to unipotent rigid fundamental groups

Applicable to smooth, projective, geometrically connected varieties

Abstract

In this short note we prove a version of Bertini's theorem for unipotent rigid fundamental groups, stating that for every smooth, projective, geometrically connected variety $X$ over an infinite perfect field $k$ of characteristic $p>0$, there exists a smooth, projective, geometrically connected curve $C\subset X$ such that the induced map on rigid fundamental groups is surjective.