Rigid toric matrix Schubert varieties

Fulton proves that the matrix Schubert variety (X-pi) over bar congruent to Y-pi x C-q can be defined via certain rank conditions encoded in the Rothe diagram of pi is an element of S-N. In the case where Y-pi := TV(sigma(pi)) is toric (with respect to a (C*)(2N-1) action), we show that it can be described as a toric (edge) ideal of a bipartite graph G(pi). We characterize the lower dimensional faces of the associated so-called edge cone sigma(pi) explicitly in terms of subgraphs of G(pi) and present a combinatorial study for the first-order deformations of Y-pi. We prove that Y-pi is rigid if and only if the three-dimensional faces of sigma(pi) are all simplicial. Moreover, we reformulate this result in terms of the Rothe diagram of pi.