On the singularity of the irreducible components of a Springer fiber in sl(n)

Details On the singularity of the irreducible components of a Springer fiber in sl(n) On the singularity of the irreducible components of a Springer fiber in sl(n)

2009-05-11

arxiv.org/abs/0905.1617v2

Selecta Math (N.S.) 16 (2010), pp. 393-418

Mathematics

Algebraic Geometry

19 pages

Scientific paper

10.1007/s00029-010-0025-z

Let ${\mathcal B}_u$ be the Springer fiber over a nilpotent endomorphism $u\in End(\mathbb{C}^n)$. Let $J(u)$ be the Jordan form of $u$ regarded as a partition of $n$. The irreducible components of ${\mathcal B}_u$ are all of the same dimension. They are labelled by Young tableaux of shape $J(u)$. We study the question of singularity of the components of ${\mathcal B}_u$ and show that all the components of ${\mathcal B}_u$ are nonsingular if and only if $J(u)\in\{(\lambda,1,1,...), (\lambda_1,\lambda_2), (\lambda_1,\lambda_2,1), (2,2,2)\}$.

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