On the normality of Schubert varieties: remaining cases in positive characteristic

Thomas J. Haines, Timo Richarz

Published 2018-06-28Version 1

Schubert varieties in classical flag varieties are known to be normal by the work of Ramanan-Ramanathan, Seshadri, Anderson and Mehta-Srinivas. The normality of Schubert varieties in affine flag varieties in characteristic zero was proved by Kumar, Mathieu and Littelmann in the Kac-Moody setting, and in positive characteristic $p > 0$ for simply connected split groups by Faltings. Pappas-Rapoport generalized Faltings' result to show that Schubert varieties are normal whenever $p$ does not divide $|\pi_1(G_{\textrm{der}})|$. This covers many cases, but the case of $\text{PGL}_n$ for $p \mid n$ is not covered, for instance. In this article, we use a variation of arguments of Faltings and Pappas-Rapoport to handle general Chevalley groups over $\mathbb{Z}$. In addition, for general Chevalley groups $G$ and parahoric subgroups $\mathcal{P} \subset LG$, we determine exactly when the partial affine flag variety $\text{Fl}_{\mathcal{P}} \otimes_{\mathbb{Z}}\mathbb{F}_p$ is reduced, and for semi-simple Chevalley groups we determine the locus in $\text{Spec}(\mathbb{Z})$ over which $\text{Fl}_{\mathcal{P}}$ is ind-flat.