Unstable loci in flag varieties and variation of quotients
Abstract
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 $X$. The main result is an explicit description of the associated unstable locus in dependence of $\mathcal L$, as well as a combinatorial 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 $q$ form a convex polyhedral cone. We also give 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 $\overline{\rm Eff}(Y_C)$ correspond to the GIT-chambers of the $\hat G$-ample cone of $X$. Moreover, all rational contractions $f: Y_{C} --\to Y'$ to normal projective varieties $Y'$ 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 $k$.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2016
- DOI:
- 10.48550/arXiv.1607.04231
- arXiv:
- arXiv:1607.04231
- Bibcode:
- 2016arXiv160704231S
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Algebraic Geometry
- E-Print:
- Second version: substantial generalization of results, and new structure theorem for the G-ample cone. 32 pages