Stratified bundles on the Hilbert Scheme of $n$ points

TL;DR

This paper computes the fundamental group scheme of the Hilbert scheme of points on a smooth projective surface over an algebraically closed field of characteristic greater than 3, using stratified bundles and Tannakian categories.

Contribution

It provides the first computation of the fundamental group scheme for Hilbert schemes of points on surfaces in positive characteristic.

Findings

Explicit computation of the fundamental group scheme for $S^{[n]}$

Extension of stratified bundle theory to Hilbert schemes

Insights into the algebraic fundamental group in positive characteristic

Abstract

Let $k$ be an algebraically closed field of characteristic $p > 3$ and $S$ be a smooth projective surface over $k$ with $k$-rational point $x$. For $n \geq 2$, let $S^{[n]}$ denote the Hilbert scheme of $n$ points on $S$. In this note, we compute the fundamental group scheme $\pi^{\textrm{alg}}(S^{[n]}, \tilde{nx})$ defined by the Tannakian category of stratified bundles on $S^{[n]}$.