Fundamental group schemes of Hilbert scheme of n points on a smooth projective surface

Stratified bundles on the Hilbert Scheme of $n$ points

“…, n r ). Let S We recall some basic properties below which we will need later (we refer to [Fog68], [PS20] for details).…”

“…Let S be a smooth projective surface over k. For n ≥ 2, let S [n] denote the Hilbert scheme n points on S. It is well known that S [n] is a smooth projective variety of dimension 2n. In [PS20], the authors show that for char k > 3 and n ≥ 2, there is an isomorphism of affine group schemes over k π † (S, x) ab → π † (S [n] , ñx) where † = S, N or ét.…”

“…Then B is a finite A module and B Sn = A. Lemma 6.1. When char k > n 1 , any S n -equivariant surjective B-module homomorphism f : M → N of finitely generated B modules descends to surjective A-module homomorphism of the S n -invariants M Sn → N SnThis allows us to prove the following analogue of Proposition 5.3.6 in[PS20].…”