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

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 $J(u)\in{(\lambda,1,1,…), (\lambda_1,\lambda_2), (\lambda_1,\lambda_2,1), (2,2,2)}$.