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

摘要

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