A note on certain Tannakian group schemes

TL;DR

This paper investigates properties of the F-fundamental group scheme in algebraic geometry, establishing its birational invariance, relation to the Nori fundamental group scheme, and behavior for elliptic curves in positive characteristic.

Contribution

It proves the birational invariance of the F-fundamental group scheme and clarifies its relationship with the Nori fundamental group scheme, especially for elliptic curves.

Findings

F-fundamental group scheme is birational invariant.

It is a quotient of the Nori fundamental group scheme.

For elliptic curves, both group schemes coincide.

Abstract

In this note, we prove that the F-fundamental group scheme is birational invariant for smooth projective varieties. We prove that the F-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the F-fundamental group scheme and the Nori fundamental group scheme coincides. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic curves, unlike in characteristic zero.