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 $k$.

ON TORUS ACTIONS OF HIGHER COMPLEXITY

We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of ‘general arrangement varieties’ and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.

A Remark on Rationally Connected Varieties and Mori Dream Spaces

In this short note, we show that a construction by Ottem provides an example of a rationally connected variety that is not birationally equivalent to a Mori dream space with terminal singularities.