Bott-Samelson-Demazure-Hansen Varieties for Projective Homogeneous Varieties with Nonreduced Stabilizers

TL;DR

This paper investigates the geometric properties of Schubert and BSDH varieties for projective homogeneous varieties with nonreduced stabilizers over fields of positive characteristic, revealing non-normality and Frobenius semisimplicity of certain morphisms.

Contribution

It provides the first detailed analysis of BSDH varieties with nonreduced stabilizers, including their non-normality and the behavior of convolution morphisms in positive characteristic.

Findings

Schubert and BSDH varieties are generally not normal.

Projection maps have nonreduced fibers at closed points.

Convolution morphisms satisfy the decomposition theorem and are Frobenius semisimple.

Abstract

Over a field of positive characteristic, a semisimple algebraic group $G$ may have some nonreduced parabolic subgroup $P$. In this paper, we study the Schubert and Bott-Samelson-Demazure-Hansen (BSDH) varieties of $G/P$, with $P$ nonreduced, when the base field is perfect. It is shown that in general the Schubert and BSDH varieties of such a $G/P$ are not normal, and the projection of the BSDH variety onto the Schubert variety has nonreduced fibers at closed points. When the base field is finite, the generalized convolution morphisms between BSDH varieties are also studied. It is shown that the decomposition theorem holds for such morphisms, and the pushforward of intersection complexes by such morphisms are Frobenius semisimple.