Singular varieties and infinitesimal non-commutative Witt vectors

TL;DR

This paper describes the group scheme (X) associated with a projective variety X over an algebraically closed field, using non-commutative Witt group schemes, especially for non-normal varieties obtained by pinching.

Contribution

It provides an explicit description of (X) via amalgamated products of non-commutative local group schemes, advancing understanding of Nori's fundamental group scheme for certain singular varieties.

Findings

(X) can be described through amalgamated products of non-commutative Witt group schemes.

The approach applies to non-normal varieties obtained by pinching a simply connected variety.

The work offers new explicit examples of local or height-specific fundamental group schemes.

Abstract

Given a projective variety $X$ over an algebraically closed field $k$, M. V. Nori introduced in 1976 a group scheme $\pi(X)$ which accounts for principal bundles $P\to X$ with finite structure, obtaining in this way an amplification the etale fundamental group. One drawback of this theory is that it is quite difficult to arrive at an explicit description of $\pi(X)$, whenever it does not vanish altogether. To wit, there are no known non-trivial examples in the literature where $\pi(X)$ is local, or local of some given height, etc. In this paper we obtain a description of $\pi(X)$ through amalgamated products of certain non-commutative local group schemes - we called them infinitesimal non-commutative Witt group schemes - in the case where $X$ is a non-normal variety obtained by pinching a simply connected one.