Singularities of closures of spherical $B$-conjugacy classes of   nilpotent orbits
  We prove that for a simply laced group, the closure of the Borel conjugacy class of any nilpotent element of height $2$ in its conjugacy class is normal and admits a rational resolution. We extend this, using Frobenius splitting techniques, to the closure in the whole Lie algebra if either the group has type $A$ or the element has rank $2$. 
Martin Bender, Nicolas Perrin
math.AG math.RT