On singularities in B-orbit closures of 2-nilpotent matrices

Abstract

This paper deals with singularities of closures of $2$-nilpotent Borel conjugacy classes in either a ${\text{GL}}_n$-conjugacy class or in the nilpotent cone of ${\text{GL}}_n$. In the latter case we construct a resolution of singularities, in the former we show that singularities are rational by applying a result of M. Brion. We reason why this generalizes the result of N. Perrin and E. Smirnov on the rationality of singularities of Springer fiber components in the two-column case. In the case of Borel orbit closures being contained in orbital varieties, we give an alternative version of L. Fresse's recent singularity criterion.

30 pages, several mistakes corrected. Comments are welcome

References in corpus (2)

• Spherical homogeneous spaces of minimal rank
• Springer fiber components in the two columns case for types A and D are normal