On the centralizer of the sum of commuting nilpotent elements

Let X and Y be commuting nilpotent K-endomorphisms of a vector space V, where K is a field of characteristic p >= 0. If F = K (t) is the field of rational functions on the projective line P-/K(1) I consider the K (t)-endomorphism A = X + t Y of V. If p = 0, or if A(p-1) = 0, we show here that X and Y are tangent to the unipotent radical of the centralizer of A in GL(V). For all geometric points (a : b) of a suitable open subset of P-1, it follows that X and Y are tangent to the unipotent radical of the centralizer of aX + bY. This answers a question of J. Pevtsova. (c) 2005 Elsevier B.V. All rights reserved.