  We prove that the nilpotent commuting variety of a reductive Lie algebra over
an algebraically closed field of good characteristic is equidimensional. In
characteristic zero, this confirms a conjecture of Vladimir Baranovsky. As a
by-product, we obtain tat the punctual (local) Hilbert scheme parametrising the
ideals of colength $n$ in $k[[X,Y]]$ is irreducible over any algebraically
closed field $k$.
