Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic

Abstract

Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}\oplus\mathfrak{g}_{\bar 1}$ be a basic classical Lie superalgebra over an algebraically closed field $\textbf{k}$ of characteristic $p>2$. Denote by $\mathcal{Z}$ the center of the universal enveloping algebra $U(\mathfrak{g})$. Then $\mathcal{Z}$ turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction $\text{Frac}(\mathcal{Z})$ is isomorphic to $\text{Frac}(\mathfrak{Z})$ for the center $\mathfrak{Z}$ of $U(\mathfrak{g}_{\bar 0})$. Consequently, both Zassenhaus varieties for $\mathfrak{g}$ and $\mathfrak{g}_{\bar 0}$ are birationally equivalent via a subalgebra $\widetilde{mathcal{Z}}\subset\mathcal{Z}$, and $\text{Spec}(\mathcal{Z})$ is rational under the standard hypotheses.