  We prove two results on the defining ideals of certain varieties of matrices.
Let us fix two positive integers r, e. Let M(r) be the set of r x r matrices
over a field K. We consider the closed subscheme of the nilpotent variety of
M(r) over K defined by the conditions char_A(T)=T^r, A^e=0. We prove that when
the characteristic of K is zero this scheme is reduced. Also for e=2 we prove
that this scheme is reduced over a field K of arbitrary characteristic.
  These results were motivated by the questions of G. Pappas and M. Rapoport
(compare their paper ''Local models in the ramified case I. The EL-case",
math.AG/0006222) and give answers to some of their conjectures.
