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

Let $k$ be an algebraically closed field of characteristic $p>3$. Let $X$ be an irreducible smooth projective surface over $k$. Fix an integer $n\ge 1$ and let ${\mathcal{H}ilb}_X^n$ 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 $\mathcal{H}{ilb}_X^n$.