Smoothness of moduli space of stable torsion-free sheaves with fixed determinant in mixed characteristic

Abstract

Let $R$ be a complete discrete valuation ring with fraction field of characteristic $0$ and algebraically closed residue field of characteristic $p>0$. Let $X_R \to \mathrm{Spec}(R)$ be a smooth projective morphism of relative dimension $1$. We prove that, given a line bundle $\mathcal{L}_R$ the moduli space of Gieseker stable torsion-free sheaves of rank $r\geq 2$ over $X_R$, with determinant $\mathcal{L}_R$, is smooth over $R$.