Andras Lorincz : On the collapsing of homogeneous bundles

Posted by Wenliang Zhang , part of the Commutative Algebra Seminar.

At

March 6, 2023, 3 p.m.

In

636 SEO

Abstract

I present results on the geometry of equivariant, proper maps from homogeneous bundles over flag varieties, called collapsing maps. Kempf showed that, provided the bundle is completely reducible, the image of a collapsing has rational singularities in characteristic zero. We extend this to positive characteristics showing that such an image is strongly F-regular if its coordinate ring has a good filtration, and give criteria for the existence of the latter. We further show that the restrictions of such collapsing maps to Schubert varieties are F-rational in positive characteristic and have rational singularities in characteristic zero. These results give a uniform, characteristic-free approach for the study of the geometry of some remarkable varieties, such as: multicones over Schubert varieties, various determinantal varieties in spaces of matrices, varieties of complexes, subspace varieties, higher rank varieties.