On embeddings of certain spherical homogeneous spaces in prime characteristic

Abstract

Let \( \mathcal{G} \) be a reductive group over an algebraically closed field of characteristic p > 0. We study embeddings of homogeneous \( \mathcal{G} \)-spaces that are induced from the G × G-space G, G a suitable reductive group, along a parabolic subgroup of \( \mathcal{G} \). We give explicit formulas for the canonical divisors and for the divisors of B-semi-invariant functions. Furthermore, we show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen–Macaulay. Examples are the G × G-orbits in normal reductive monoids with unit group G. Further examples are the open \( \mathcal{G} \)-orbits of the well known determinantal varieties and the varieties of (circular) complexes. Finally, we study the Gorenstein property for the varieties of circular complexes and for a related reductive monoid.