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Varieties of reductions for $gl\_n$

Atanas Iliev·Laurent Manivel
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Abstract

We study the varieties of reductions associated to the variety of rank one matrices in $\fgl\_n$. These varieties are defined as natural compactifications of the different ways to write the identity matrix as a sum of $n$ rank one matrices. Equivalently, they compactify the quotient of $PGL\_n$ by the normalizer of a maximal torus. In particular, we prove that for $n=4$ we get a 12-dimensional Fano variety with Picard number one, index 3, and canonical singularities.

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Atanas Iliev, Laurent Manivel. 2005-01-20. Varieties of reductions for $gl\_n$. https://arxiv.org/abs/math/0501329

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