Jordan forms for mutually annihilating nilpotent pairs

We consider pairs of n × n commuting matrices over an algebraically closed field F . For n, a, b (all at least 2) let V(n, a, b) be the variety of all pairs (A,B) of commuting nilpotent matrices such that AB = BA = A = B = 0. In [14] Schröer classified the irreducible components of V(n, a, b) and thus answered a question stated by Kraft [9, p. 201] (see also [3] and [10]). If μ = (μ1, μ2, . . . , μt) is a partition of n then we denote by Oμ the conjugacy class of all nilpotent matrices such that the sizes of Jordan blocks in its Jordan canonical form are equal to μ1, μ2, . . . , μt. Let μ = (μ1, μ2, . . . , μt) and ν = (ν1, ν2, . . . , νs) be partitions of n such that μ1 ≤ a and ν1 ≤ b and let π1(A,B) = A and π2(A,B) = B for (A,B) ∈ V(n, a, b) be the projection maps. Schröer [14, p. 398] noted that the intersection of the fibers π 1 (Oμ) ∩ π −1 2 (Oν) is not very well-behaved for different reasons. It might be empty or reducible and the closure of the intersection of fibers is in general not a union of such intersections. In this paper we answer the question for which pairs of partitions (μ, ν) the intersection π 1 (Oμ) ∩ π −1 2 (Oν) is nonempty. If μ = (μ1, μ2, . . . , μt, 1 ), where μt ≥ 2, is a fixed partition, then partitions ν, such that π 1 (Oμ) ∩ π −1 2 (Oν) is nonempty, are of the form