Rigid toric matrix Schubert varieties

Abstract Fulton proves that the matrix Schubert variety $$\overline{X_{\pi }} \cong Y_{\pi } \times \mathbb {C}^q$$ X π ¯ ≅ Y × C q can be defined via certain rank conditions encoded in Rothe diagram of $$\pi \in S_N$$ ∈ S N . In case where $$Y_{\pi }:={{\,\textrm{TV}\,}}(\sigma _{\pi })$$ : = TV ( σ ) is toric (with respect to a $$(\mathbb {C}^*)^{2N-1}$$ ∗ 2 - 1 action), we show it described as (edge) ideal bipartite graph $$G^{\pi }$$ G We characterize lower dimensional faces associated so-called edge cone $$\sigma explicitly terms subgraphs and present combinatorial study for first-order deformations prove rigid if only three-dimensional are all simplicial. Moreover, reformulate this result $$