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 pgreater-or-equal, slanted0. If F=K(t) is the field of rational functions on the projective line image, consider the K(t)-endomorphism A=X+tY of V. If p=0, or if Ap-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 P1, 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.