# Fundamental Group Schemes of $n$-fold Symmetric Product of a Smooth Projective Curve

## TL;DR

This paper computes the $S$-fundamental and Nori's fundamental group schemes of the $n$-fold symmetric product of a smooth projective curve over an algebraically closed field of characteristic p > 0.

## Contribution

It provides the first explicit determination of the fundamental group schemes of symmetric products of curves in positive characteristic.

## Key findings

- Determined the $S$-fundamental group scheme of $S^n(X)$.
- Computed Nori's fundamental group scheme for $S^n(X)$.
- Extended understanding of fundamental group schemes in algebraic geometry.

## Abstract

Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this article we find the $S$-fundamental group scheme and Nori's fundamental group scheme of $S^n(X)$.