Intersections of closures of table loci for stable two-part partitions

Determine the intersections of the closures of the loci W_{Q,k,l} and W_{Q,k',l'} inside the nilpotent commutator N_{J_Q} for stable two-part partitions Q = (u, u − r) with r ≥ 2, in cases where neither locus contains the other. Here N_{J_Q} denotes the linear space of nilpotent matrices commuting with a Jordan matrix J_Q of Jordan type Q, and W_{Q,k,l} denotes the locally closed subset where the Jordan type P_B equals the partition P_{k,l}(Q) labeling the (k,l) entry in the box D^{-1}(Q) associated to Q, with indices 1 ≤ k ≤ r − 1 and 1 ≤ l ≤ u − r.

Background

For stable partitions Q = (u, u − r), the paper determines explicit complete intersection equations E_{k,l} defining the closures of the loci W_{Q,k,l} within the nilpotent commutator N_{J_Q}, and shows these closures are irreducible of codimension k + l − 2. Corollary 3.8 establishes the dominance order among these loci.

Despite these structural results, the authors note that understanding the intersections of closures of different table loci when there is no containment remains unresolved. Example 4.1 illustrates that such intersections can be reducible and that specialization among table loci is not monotone in row or column indices, emphasizing the complexity of the intersection behavior.