On the singular locus of certain subvarieties of Springer fibers

Let $x\in\mathrm{End}(\mathbb{K}^n)$ be an endomorphism such that $x^2=0$(where $\mathbb{K}$ is an algebraically closed field).The corresponding Springer fiber $\mathcal{F}_x$ is the algebraic variety of $x$-stable complete flags. In the present case, $\mathcal{F}_x$ has a suitable decomposition into a finite number of orbits under the action of the centralizer of $x$.The closures of these orbits may be singular. In this paper, we give a combinatorial description of the singular locus of the orbit closures. In particular, we deduce a description of the singular locus of the irreducible components of $\mathcal{F}_x$.

Keywords

flag varieties, Springer fibers, singular locus, link patterns

2010 Mathematics Subject Classification

14L30, 14M15, 17B08

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Published 27 December 2012