# Fundamental Group Schemes of Hilbert Scheme of $n$ Points on a Smooth Projective Surface

## TL;DR

This paper investigates the fundamental group schemes of the Hilbert scheme of points on a smooth projective surface over an algebraically closed field of characteristic greater than 3, extending understanding of their algebraic fundamental groups.

## Contribution

It determines the $S$-fundamental group scheme and Nori's fundamental group scheme of the Hilbert scheme of points on a smooth projective surface.

## Key findings

- Computed the $S$-fundamental group scheme of the Hilbert scheme.
- Computed Nori's fundamental group scheme of the Hilbert scheme.
- Extended fundamental group scheme theory to Hilbert schemes of points.

## Abstract

Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $X$ be an irreducible smooth projective surface over $k$. Fix an integer $n \geq 1$ and let ${\mathcal{H}{\it ilb}}_X^n$ be the Hilbert scheme parameterizing effective $0$-cycles of length $n$ on $X$. The aim of the present article is to find the $S$-fundamental group scheme and Nori's fundamental group scheme of the Hilbert scheme $\mathcal{H}{\it ilb}_X^n$.