Unstable Loci in Flag Varieties and Variation of Quotients

We consider the action of a semisimple subgroup $\hat{G}$ of a semisimple complex group $G$ on the flag variety $X=G/B$ and the linearizations of this action by line bundles $\mathcal L$ on $X$. We give an explicit description of the associated unstable locus in dependence of $\mathcal L$, as well as a formula for its (co)dimension. We observe that the codimension is equal to 1 on the regular boundary of the $\hat{G}$-ample cone and grows towards the interior in steps by 1, in a way that the line bundles with unstable locus of codimension at least $q$ form a convex polyhedral cone. We also give a description and a recursive algorithm for determining all GIT-classes in the $\hat{G}$-ample cone of $X$. As an application, we give conditions ensuring the existence of GIT-classes $C$ with an unstable locus of codimension at least two and which moreover yield geometric GIT quotients. Such quotients $Y_C$ reflect global information on $\hat{G}$-invariants. They are always Mori dream spaces, and the Mori chambers of the pseudoeffective cone $\overline{\textrm{Eff}}(Y_C)$ correspond to the GIT chambers of the $\hat{G}$-ample cone of $X$. Moreover, all rational contractions $f: Y_{C} \ \scriptsize{-}\scriptsize{-}{\scriptsize{-}\kern-5pt\scriptsize{>}}\ Y^{\prime}$ to normal projective varieties $Y^{\prime}$ are induced by GIT from linearizations of the action of $\hat{G}$ on $X$. In particular, this is shown to hold for a diagonal embedding $\hat{G} \hookrightarrow (\hat{G})^k$, with sufficiently large $k$.

Groups with prescribed automorphism group: A clarification

In Theorems 1 and 2 of [] necessary and sufficient conditions were given for a group G to have a finite automorphism group Aut G and a semisimple subgroup of central automorphisms AutcG. Recently it occurred to us, as a result of conversations with Ursula Webb, that these conditions could be stated in a much simpler and clearer form. Our purpose here is to record this reformulation. For an explanation ofterminology and notation we refer the reader to [1].