The tame fundamental group schemes of curves in positive characteristic

TL;DR

This paper investigates the structure of tame fundamental group schemes of smooth curves over algebraically closed fields of positive characteristic, revealing their dependence on the curves and their relation to numerical invariants.

Contribution

It develops the theory of cospecialization maps for tame fundamental group schemes and shows how these schemes encode numerical invariants of the curves.

Findings

Tame fundamental group schemes depend heavily on the specific curve.

Numerical invariants of curves can be reconstructed from their tame fundamental group schemes.

The paper establishes the theory of cospecialization maps for these schemes.

Abstract

The tame fundamental group scheme for an algebraic variety is the maximal linearly reductive quotient of Nori's fundamental group scheme. In this paper, we study the tame fundamental group schemes of smooth curves defined over algebraically closed fields of positive characteristic and develop the theory of cospecialization maps for them. As a result, we see that the tame fundamental group schemes heavily depend on the curves. We also see that numerical invariants of curves can be reconstructed from the tame fundamental group schemes.