On the singularity of the irreducible components of a Springer fiber in sl_n

Let B_u be the Springer fiber over a nilpotent endomorphism u ∈ End(ℂⁿ). Let J (u) be the Jordan form of u regarded as a partition of n. The irreducible components of B_u are all of the same dimension. They are labelled by Young tableaux of shape J (u). We study the question of the singularity of the components of B_u and show that all the components of B_u are nonsingular if and only if J(u) ∈ {(λ, 1, 1,...), (λ₁, λ₂), (λ₁, λ₂, 1), (2, 2, 2)}.