Rigid toric matrix Schubert varieties

Fulton proves that the matrix Schubert variety Xπ¯≅Yπ×Cq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{X_{\pi }} \cong Y_{\pi } \times \mathbb {C}^q$$\end{document} can be defined via certain rank conditions encoded in the Rothe diagram of π∈SN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi \in S_N$$\end{document}. In the case where Yπ:=TV(σπ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{\pi }:={{\,\textrm{TV}\,}}(\sigma _{\pi })$$\end{document} is toric (with respect to a (C∗)2N-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {C}^*)^{2N-1}$$\end{document} action), we show that it can be described as a toric (edge) ideal of a bipartite graph Gπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G^{\pi }$$\end{document}. We characterize the lower dimensional faces of the associated so-called edge cone σπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\pi }$$\end{document} explicitly in terms of subgraphs of Gπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G^{\pi }$$\end{document} and present a combinatorial study for the first-order deformations of Yπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{\pi }$$\end{document}. We prove that Yπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{\pi }$$\end{document} is rigid if and only if the three-dimensional faces of σπ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\pi }$$\end{document} are all simplicial. Moreover, we reformulate this result in terms of the Rothe diagram of π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document}.


Introduction
In this paper, we are studying the matrix Schubert varieties X π ∼ = Y π ×C q associated to a permutation π ∈ S N , where q is maximal possible.These varieties first appear during Fulton's study of the degeneracy loci of flagged vector bundles in [7].In [11], Knutson and Miller show that Schubert polynomials are multidegrees of matrix Schubert varieties.Moreover the matrix Schubert variety is in fact related to Schubert variety X π in the full flag manifold via the isomorphism in ( [10], Lemma A.4).The matrix Schubert varieties are normal and one can define them by certain rank conditions encoded in the Rothe diagram.Our goal is to investigate the natural restricted torus action on these varieties.Escobar and Mészáros [6] study the toric matrix Schubert varieties via understanding their moment polytope.We present a reformulation of their classification in terms of bipartite graphs.The significance of this restatement is that it allows one to study the first order deformations of the matrix Schubert variety in terms of graphs by [13].The toric varieties arising from bipartite graphs have been studied in various papers [4], [9], [12], [14] in different perspectives.We will review few aspects of this theory and bring our attention to the classification of the rigid toric matrix Schubert varieties.The toric varieties arising from graphs enable us to produce many interesting examples of rigid varieties.In fact, the first example of a rigid singularity in [8] is the cone over the Segre embedding P 2 × P 1 → P 2r+1 which is the affine toric variety associated to the complete bipartite graph K r+1,2 .Following the techniques in [1,13] for the study of deformations of toric varieties, we classify rigid toric varieties Y π in terms of bipartite graphs and Rothe diagram.
The organization of the paper is as follows.In preliminaries we present some basic facts on matrix Schubert varieties and give a brief exposition of toric varieties arising from bipartite graphs.In Section 3 we reformulate the question of classification of toric matrix Schubert varieties to bipartite graphs.We then indicate how graphs may be used to investigate the complexity of the torus action in the sense of T -varieties [2], [3].Section 4 starts with a discussion of deformation theory of toric varieties.Furthermore it provides a detailed exposition of faces of the moment (edge) cone of Y π .In Lemma 4.5, we characterize the extremal rays of the edge cone.In Proposition 4.8 and Proposition 4.11, we present conditions for extremal rays to span a two-dimensional face and a three-dimensional face respectively.Finally, we conclude the following result.
Theorem (Theorem 4.12).The toric variety Y π is rigid if and only if its moment (edge) cone has simplicial three-dimensional faces.
We translate this result to the Rothe diagram of π and restate the classification in terms of certain shapes on the diagram, in particular solely depending on the pattern of the essential set (Definition 2.4) of π.
Corollary (Corollary 4.13).Let Ess(π) = {(x i , y i ) | x k+1 < . . .< x 1 and y 1 < . . .< y k+1 } with k ≥ 3 be the essential set of the Rothe diagram of π ∈ S N .Then the toric variety Y π is rigid if and only if where m is the length and n is the width of the smallest rectangle containing L(π) from Definition 2.4.

Preliminaries
2.1.Matrix Schubert varieties.In this section, we adopt the conventions from [6] for matrix Schubert varieties.We are mainly interested in matrix Schubert varieties for their effective torus actions and deformations.The statements presented in this section can be found in [7] and [11].
Let π ∈ S N be a permutation.We denote its permutation matrix by π ∈ C N ×N as well and define it as follows: Let B denote the group of invertible lower triangular matrices and B + denote the group of invertible upper triangular N × N matrices.The product B × B + acts from left on C N ×N and its action defined as:    Remark 1.By the previous theorem, we observe that there exists no rank conditions imposed on the boxes which are not in NW(π).Thus these boxes are free in X π .Let V π ∼ = C N 2 −| NW(π)| be the projection of the matrix Schubert variety X π ⊆ C N ×N onto these free boxes.Also, we define Y π as the projection onto the boxes of L(π).Note that one obtains (a, b) ∈ dom(π) if and only if r π (a, b) = 0. Hence, X π = Y π × V π holds.In particular, by Theorem 2.5, Example 1.The essential set for the permutation π = [2143] ∈ S 4 consists of the boxes (1, 1) and (3,3).Let M = (m ij ) ∈ C 4×4 .First we note that m 11 = 0 since (1, 1) ∈ dom(π).For the boxes in L(π) one obtains the following inequality by Theorem 2.5: One obtains the ideal as generated by I := det(M (3,3) ).In particular 2.2.Edge cones of bipartite graphs.In this section, we briefly introduce the construction of the toric varieties related to bipartite graphs as in [9,13].We refer the reader to [5] for details on the toric varieties and the notations.In particular N ∼ = Z n stands for a lattice and M = Hom Z (N, Z) is its dual lattice.We denote their associated vector spaces as Let G ⊆ K m,n be a bipartite graph with edge set E(G) and vertex set V (G).One defines the edge ring as Edr(G) : Consider the following morphism: x e → t i t j with e = (i, j).
The kernel of this map is a toric ideal and it is called the edge ideal of G.The affine normal toric variety associated to bipartite graph G is Let e i denote a canonical basis element of Z m × 0 for i = 1, . . ., m and f j denote a canonical basis element of 0 × Z n for j = 1, . . ., n. Set the lattices for the associated cones of the toric variety TV(G) as ⊥ where (1, −1) := m i=1 e i − n j=1 f j .We denote their associated vector spaces as N Q and M Q .In order to distinguish the elements of these vector spaces, we denote the ones in N Q by normal brackets and the ones in M Q by square brackets.For the same reason, we denote the canonical basis elements as e i ∈ N Q and e i ∈ M Q .
Hence we obtain that the (dual) edge cone associated to TV(G) is ).We observe in Section 3 that the dual edge cone σ ∨ G is in fact isomorphic to the moment cone of a matrix Schubert variety.We use this fact in order to determine the complexity of the torus action on a matrix Schubert variety.Proposition 2.6 ([13, Proposition 2.1, Lemma 2.17]).Let G ⊆ K m,n be a bipartite graph with k connected components and m + n vertices.Then the dimension of σ Our aim is to study the first order deformations T 1 TV(G) of the affine toric variety TV(G) by [1].One can describe the T 1 TV(G) via understanding the two and three-dimensional faces of the edge cone σ G ⊆ N Q .We explain this technique briefly in Section 4.1.We first introduce terminology and notation from graph theory to describe the rays and faces of σ G in terms of subgraphs of G.In Section 4.2, we will describe the rigidity of TV(G) in terms of graphs.
).Thus for the remainder of this section, we assume that G ⊆ K m,n is connected.Definition 2.7.A nonempty subset A of V (G) is called an independent set if it contains no adjacent vertices.An independent set A V (G) is called a maximal independent set if there is no other independent set containing it.Let U 1 and U 2 be the disjoint sets of vertices of G.We say that an independent set is one-sided if it is contained either in U 1 or U 2 .In a similar way, The supporting hyperplane of the dual edge cone σ ∨ G ⊆ M Q associated to an independent set ∅ = A is defined as x i }.
Note that since no pair of vertices of an independent set A is adjacent, we obtain that A∩N (A) = ∅.
(1) A subgraph of G with the same vertex set as G is called a spanning subgraph of G.
(2) Let A ⊆ V (G) be a subset of the vertex set of G.The induced subgraph of A is defined as the subgraph of G formed from the vertices of A and all of the edges connecting pairs of these vertices.We denote it as G[A] and we adopt the convention Now, we characterize the independent sets resulting a facet of σ ∨ G .Definition 2.10.Let G[[A]] denote the subgraph of G associated to the independent set A and defined as Finally, we define the first independent sets I G of G as I G := Two-sided maximal independent sets and one-sided independent sets U i \{•} where their associated bipartite subgraph has two connected components.Note that Definition 2.10 becomes less technical for first independent sets by [13, Proposition 2.9, Lemma 2.10].Namely we obtain: Denote the set of extremal ray generators (i.e.1-dimensional faces) of σ G by σ G .Recall that there is a bijective inclusion-reversing correspondence between the faces of σ G and the faces of σ ∨ G .Given a face τ σ ∨ G , we define the dual face τ * of τ as {x ∈ σ ∨ G | x, u = 0 for all u ∈ τ }.In particular the facets of σ ∨ G are in bijection with the extremal rays of σ G .Theorem 2.11 ([13, Theorem 2.8]).There is a one-to-one correspondence between the set of extremal generators σ G .In particular, the map is given as Example 2. We consider the bipartite graph G ⊂ K 2,2 obtained by removing one edge from the complete bipartite graph.The first independent set I G since they are contained in the two-sided maximal independent set {1, 3}.Hence their associated subgraph has three connected components.The cone σ G ⊆ N Q is generated by (1, 0, 0, 0), (0, 0, 1, 0) and (0, 1, −1, 0) corresponding respectively to the associated subgraphs.

G{{1, 3}}
The next result classifies d-dimensional faces of σ G via intersecting associated subgraphs related to first independent sets.
Theorem 2.12 ([13, Theorem 2.18]).Let S ⊆ I G be a subset of d first independent sets and let Π be the bijection from Theorem 2.11.The extremal ray generators Π(S) span a face of dimension d if and only if the dimension of the dual edge cone of the spanning subgraph G[S] := A∈S G{A} is m+n−d−1, i.e.G[S] has d+1 connected components.In particular, the face is equal to H Val S ∩σ G where Val S is the degree sequence of the graph G[S] and Example 3.All pairs of extremal rays of σ G generates a two-dimensional face of σ G since the intersection of all pairs of associated subgraphs has three connected components.In particular, the twodimensional face generated by (1, 0, 0, 0) and (0, 1, −1, 0), i.e. the edge cone of G{{1}} ∩ G{{1, 3}}, is equal to

Torus action on matrix Schubert varieties in terms of graphs
We are interested in the torus action on Y π .This question has been first studied by Escobar and Mészáros in [6] where they characterize all toric varieties Y π .We reformulate this classification in terms of graphs.Moreover we approach the question of determining the dimension of the torus acting on Y π from a perspective of T -varieties.These are normal varieties with effective torus action having not necessarily a dense torus orbit.They can be considered as the generalization of toric varieties with respect to the dimension of their torus action.For more details about T -varieties, we refer to [2], [3].Definition 3.1.An affine normal variety X is called a T-variety of complexity-d if it admits an effective T torus action with dim(X) − dim(T ) = d.
The matrix Schubert varieties are normal varieties (see [11], Theorem 2.4.3.).The action of B ×B + on X π restricts to the action of T N × T N , where Since this action is not effective ((aI N , aI N ).M = M), we consider the stabilizer Stab((C * ) 2N ) of this torus action and the action of the quotient Let p be a general point in Y π which have 1 in all boxes of L(π) and 0 in others.Then the closure of the torus orbit (C * ) 2N .p is the affine toric variety associated to the so-called (C * ) 2N -moment cone (or weight cone) of Y π , denoted by Φ(Y π ).One obtains that dim(Φ(Y π )) = dim((C * ) 2N .p).Since (C * ) 2N .pand Y π are both irreducible, it suffices to examine their dimension for the complexity of the torus action on Y π .Recall that the convex polyhedral cone generated by all weights of the torus action on Y π in M R (vector space spanned by the character lattice of considered torus) is called the weight cone.Here, the weight cone of the action can be expressed as where e i denotes the canonical basis for R m × 0 and f j denotes the canonical basis for 0 × R n .Note that this cone is GL-equivalent to a dual edge cone associated to a bipartite graph.Hence one can define a bipartite graph G π ⊆ K m,n from a Rothe diagram D(π) via the following bijection: where for (a, b) ∈ E(G π ), a ∈ U 1 and b ∈ U 2 .Hence we obtain also the vertex set V (G π ).We denote the associated edge cone by σ π .By Remark 1, we conclude the following.( For the complexity zero case, i.e. toric case, we present an alternative proof with edge cones.Proof.By Proposition 3.2, we aim to characterize the case when dim(σ ∨ π ) = L (π).Note that L(π) has a skew shape.Assume that L(π) consists of k connected components with m i rows and n i columns for each i ∈ [k].This means that we investigate the bipartite graph G π ⊆ K m,n with k connected bipartite graph components G π i ⊆ K m i ,n i .By Proposition 2.6, the dimension of the cone dim(σ π ) is m + n − k.Since L(π) has k connected components, the components of L (π) for each i ∈ [k] do not share a row or a column.Therefore, we are left with proving the statement for a connected component L i (π) of L(π).The dimension of the dual edge cone of G π i is equal to |L i (π)| if and only if L i (π) has a hook shape.

Rigidity of toric matrix Schubert varieties
This section is devoted to the study of the detailed structure of the edge cone σ π for matrix Schubert varieties X π where Y π = TV(G π ) is toric.Note that these matrix Schubert varieties are called toric matrix Schubert varieties in [6] and we adopt this convention.First, we explain briefly the combinatorial techniques for the first order deformations of toric varieties.By studying the first independent sets of G π and the two and three-dimensional faces of σ π , we present the conditions for rigidity of toric matrix Schubert varieties.By Remark 2 and since we investigate rigidity, we can assume that L(π) is connected.Throughout this section, the connected bipartite graph G π ⊆ K m,n denotes the associated bipartite graph of L(π) which was constructed in Section 3.

Deformations of toric varieties.
A deformation of an affine algebraic variety X 0 is a flat morphism π : X −→ S with 0 ∈ S such that π −1 (0) = X 0 , i.e. we have the following commutative diagram.
The variety X is called the total space and S is called the base space of the deformation.Let π : X −→ S and π : X −→ S be two deformations of X 0 .We say that two deformations are isomorphic if there exists a map φ : X −→ X over S inducing the identity on X 0 .Let A be an Artin ring and let S = Spec(A).One has a contravariant functor Def X 0 such that Def X 0 (A) is the set of deformations of X 0 over S modulo isomorphisms.
Definition 4.1.The map π is called a first order deformation of The variety X 0 is called rigid if T 1 X 0 = 0.This implies that a rigid variety X 0 has no nontrivial infinitesimal deformations.This means that every deformation π ∈ Def X 0 (A) over S = Spec(A) is trivial i.e. isomorphic to the trivial deformation X 0 × S −→ S.
For the case where X 0 = Spec(C[σ ∨ ∩ M ]) is an affine normal toric variety, we introduce the techniques which are developed in [1] in order to investigate the C-vector space T 1 X 0 .The deformation space T 1 X 0 is multigraded by the lattice elements of M , i.e.T 1 We first set some definitions in order to describe the homogeneous part T 1 X 0 (−R).Definition 4.2.Let us call R ∈ M a deformation degree and let σ ⊆ N Q be generated by the extremal ray generators a 1 , . . ., a k .We define the following affine space The vector ¯ ∈ {0, ±1} α is called a sign vector assigned to each two-dimensional compact face of Q(R) defined as i = ±1, if d i is an edge of 0 such that i∈[α] i d i = 0, i.e the oriented edges i d i form a cycle along the edges of .We choose one of both possibilities for the sign of .
(ii) For every deformation degree R ∈ M , the related vector space is defined as The toric variety TV(G) associated to a bipartite graph G ⊆ K m,n is smooth in codimension 2 ([13], Theorem 4.5).Hence we introduce the formula for this special case.
. Moreover, it is built by those t's satisfying t ij = t jk where a j is a non-lattice common vertex in Q(R) of the edges d ij = a i a j and d jk = a j a k .Remark 3. The following two cases of Q(R) in Figure 3 will appear often while we study T 1 TV(G) (−R).Let us interpret these cases with the previous result.• Let 1 , 2 Q(R) be the compact 2-faces sharing the edge d 3 .We choose the sign vectors 1 = (1, 1, 1, 0, 0) and 2 = (0, 0, 1, 1, 1).Suppose that t = (t 1 , t 2 , t 3 , t 4 , t 5 ) ∈ V (R).We observe that t 1 = t 2 = t 3 for 2-face 1 and t 3 = t 4 = t 5 for 2-face 2 .
• Let 1 , 2 Q(R) be the compact 2-faces connected by the vertex a j .As in the previous case we obtain that t 1 = t 2 = t 3 and t 4 = t 5 = t 6 .By Theorem 4.4, if a j is a non-lattice vertex, then we obtain t 3 = t 4 .

4.2.
Faces of the edge cone σ π of toric variety Y π .In order to study the rigidity of Y π = TV(σ π ) with Theorem 4.4, we investigate the face structure of the edge cone σ π more closely.We consider three types of first independent sets with following notations: the one-sided first independent sets A = U 1 \{•}, B = U 2 \{•} and two-sided (maximal) first independent sets C = C 1 C 2 .We label the essential boxes from the bottom of the diagram starting with (x 1 , y 1 ) to the top ending with (x k+1 , y k+1 ) i.e. we have x k+1 < . . .< x 1 and y 1 < . . .< y k+1 .Lemma 4.5.For any permutation π ∈ S N , (1) The one-sided first independent sets of G π are U i \{u i } for all u i ∈ U i and for i = 1, 2.
(2) The two-sided first independent sets are the maximal two-sided independent sets of G π .
Proof.By Theorem 3.3, L (π) is a hook.The boxes of L(π) form a shape of a Ferrer diagram, i.e. we have λ 1 ≥ . . .≥ λ t where λ i denotes the number of boxes at ith row of L(π).Consider the smallest rectangle containing L(π) of length m and of width n.The removed edges of the bipartite graph G π ⊆ K m,n are linked with the free boxes of X π in the rectangle.Let (x i , y i ) ∈ Ess(π), equivalently let (x i , y i ) ∈ E(G π ).Then one obtains naturally that there exists a two-sided maximal independent set C = C 1 C 2 = {x i + 1, . . ., m} {y i−1 + 1, . . ., n} where (x i−1 , y i−1 ) ∈ Ess(π) with x i−1 > x i and y i−1 < y i .Then the neighbor sets are ] also form a shape of a Ferrer diagram and G{C} has two connected components.In particular, U i \{u i } cannot be contained in a two-sided independent set.Suppose that G{U i \{u i }} has more than three components.Then as in [13, Proposition 2.13], there exist two-sided first independent sets C i ∈ I G such that C i 1 = U i \{u i } which is not possible.Proof.Consider again the smallest rectangle containing L(π) of a length m and of a width n.If there exists only one essential set of π, then G π = K m,n .Assume that there are more than one essential box.Let (x j , y j ) and (x i , y i ) be two essential boxes with j < i, x j > x i and y j < y i .By Lemma 4.5, we obtain two first independent sets C = {x i + 1, . . ., m} {y i−1 + 1, . . ., n} and C = {x j + 1, . . ., m} {y j−1 + 1, . . ., n} of G π .We infer that Example 6.The boxes of L(π) for the toric variety Y π is presented in Figure 4.The blue boxes are removed edges between some vertex sets C 1 ⊂ U 1 and C 2 ⊂ U 2 .We observe that C := C 1 C 2 is a maximal independent set.In particular, the orange color represents the edges of the induced subgraph G[C 1 N (C 1 )] and the purple color represents the edges of the induced subgraph G[C 2 N (C 2 )].The crossed boxes are the boxes of the essential set Ess(π).The boxes with a dot form the shape of a hook and these are the boxes of L (π).
G π and G{C} for a matrix Schubert variety X π Let us first identify the cases where there is one or there are two essential boxes.Proof.It follows from [13,Theorem 4.3,4.6].
From now on, we assume that | Ess(π)| ≥ 3.This means that we consider the associated connected bipartite graph G π K m,n with m, n ≥ 4. We denote by G π the set of tuples of first independent sets forming a d-dimensional face of σ G .Let σ (d) G π be the set of d-dimensional faces of σ π .Recall the classification of d-dimensional faces of an edge cone in Theorem 2.12 for a subset S = {A (1) , . . ., A (d) } ⊆ I G of d first independent sets.Let Π be the isomorphism from Theorem 2.11.Then we have (1) , . . ., A (d) ) → (Π(A (1) ), . . ., G π if and only if there exists a first independent set G π if and only C 1 = {i} or there exists C ∈ I (1) Proof.
(1) Suppose that there exist a pair (A, B) / ∈ I G π .Consider the intersection subgraph G{A} ∩ G{B} and assume that it has isolated vertices other than {i, j}.Consider the isolated vertices in U 1 \{i}.This means that there exists a two-sided independent set consisting of these isolated vertices and B, which is impossible, since B ∈ I G π .Now assume that G{A} ∩ G{B} consists of the isolated vertices {i, j} and k ≥ 2 connected bipartite graphs G i .Let the vertex set of G i consist of V i U 1 and W i U 2 .Since B ∈ I (1) G π , there exist an edge (i, w i ) ∈ E(G π ) for each i ∈ [k] where w i ∈ W i .Symmetrically, since A ∈ I (1) G π , there exist an edge (j, v i ) ∈ E(G π ) for each i ∈ [k] where v i ∈ V i .However, then for I [k], we obtain the two-sided maximal independent sets of form i∈I V i (B\( i∈I W i ) which contradicts the construction of G π .
(2) Let (x j , y j ) and (x i , y i ) be two essential boxes with x j > x i and y j < y i , associated to two first independent sets C and C in I (1) is connected.We observe that the edges of this graph are represented by the square with vertices (x i + 1, y j−1 + 1), (x i + 1, y i−1 ), (x j , y j−1 + 1), and (x j , y i−1 ), intersected with the diagram D(π).This intersection is also a Ferrer diagram and connected.
(3) Consider the intersection subgraph G{A} ∩ G{A }.Assume that it has only {i, i } U 1 as isolated vertices and k connected bipartite graphs.Then, as in case 1, there exist first independent sets C, C with C 1 ∩ C 1 = ∅, which is impossible by Lemma 4.6.Assume that it has the isolated vertices {i, i } U 1 and C 2 U 2 with |C 2 | ≤ n − 2. Then C := U 1 \{i, i } C 2 is maximal and thus a first independent set.
Assume that e 1 , e 2 , and e 3 are vertices in Q(R) for some deformation degree π be an extremal ray.Since (a, e i , e j ) spans a 3-face of σ π for every i, j ∈ [3] and i = j, we are left with showing that there exists no such a ∈ Q(R).However, even though we have that R i ≤ 0, for every i ∈ 3. Suppose that (e 1 , e 2 ) does not span a 2-face and e 1 and e 2 are in Q(R) for some deformation degree R ∈ M .Then there exists a first independent set C = C 1 C 2 ∈ I there can be at most one non 2-face pair say (f 1 , f 2 ).However, the other triples of type (A, B, B ) not containing both U 2 \{1} and U 2 \{2} form 3-faces. Suppose now that c i ∈ Q(R) is a lattice vertex.We can assume that there exists only one such extremal ray c i , since any triple of type (C, C , U 1 \{1}) and (C, C , U 1 \{2}) form 3-faces. Moreover there exists at most one f j such that (f j , c i ) do not span a two-dimensional face.Hence we obtain that V (R)/C(1, . . ., 1) = 0 for this deformation degree R ∈ M .It leaves us to check the case where k = 2.In this case, if the pair {f 1 , f 2 } do not span a 2-face σ π , then there exists a first independent set Then the only other vertex in Q(R) is c and it is not a lattice vertex.Furthermore, (e i , f j , c) spans three-dimensional faces of σ π for i ∈ [2] and j ∈ [2].Last, assume that (f j 1 , f j 2 ) spans a 2-face of σ G π .As in the case where k ≥ 3, it is enough to check the cases for only one vertex c i in Q(R).There exists at most one non 2-face pair containing c i , say (f j 1 , c).But then (c j , f j 2 , e 1 ) is a 3-face of σ G π .4. Lastly, suppose that {c, e i } does not span a 2-face and c and e i are in Q(R) for some deformation degree R ∈ M .Remark here that we excluded the cases where there exist non-simplicial threedimensional faces.This means c and e i forms 2-faces with each extremal ray of σ π .Assume that there exist more than three vertices in Q(R) other than c and e i .We examined the cases where non 3-face (e 1 , e 2 , e 3 ) appears and where non 2-face (e 1 , e 2 ) appears in Q(R).Therefore we assume that there exists another non 2-face pair, say (c * , e j ).But, since c * and e j also forms 2-faces with each extremal ray of σ π , it is enough to check the cases where there exist less than five vertices in Q(R).Let us first consider the case where there exist exactly two more vertices in Q(R) other than c and e i .We first start with the non 2-face pair (A, C) where C 1 = {m} and A = U 1 \{m}.Then there exists a non-lattice vertex j ∈ Q(R) where j ∈ U 2 \C 2 .We observe that there exists no other first independent set C ∈ I (1) G π such that C 2 C 2 .Therefore it is impossible that there exists another non 2-face pair containing c .
In the other case where (c, e i ) does not span a 2-face, there exists an extremal ray, say c such that c = e i + c − j∈C 2 \C 2 f j .The vertex c is in Q(R), unless there exists f j ∈ Q(R) where j ∈ C 2 \C 2 .This vertex cannot be f j with {j} = C 2 \C 2 , because then (c, c , e i , f j ) spans a 3-face.Hence c is one of these two vertices.It remains to check the case where other vertex is e i−1 .Then, there exists a first independent set C ∈ I (1) G π .We have that c / ∈ Q(R) if and only if there exists f j with j ∈ C 2 \C 2 , by the same reasoning as before.Lastly, assume that there exists only one lattice vertex in Q(R) other than c and e i .We observe that c is a lattice vertex of Q(R) if there exist some f j ∈ Q(R) where j ∈ C 2 \C 2 .Therefore we assume that this lattice vertex is f j for some j ∈ [n].In order to obtain R, c = 0, we must have {j} = C 2 \C 2 , but this implies that (c, c , e i , f j ) is a 3-face of σ π .
We interpret the rigidity of Y π by giving certain conditions on the Rothe diagram.Therefore Y π is not rigid.On the other hand, we observe the toric variety in Example 6 is rigid.

Definition 2 . 2 .
,b) ∈ C a×b be the a × b matrix located at the upper left corner of M ∈ C N ×N , where 1 ≤ a ≤ N and 1 ≤ b ≤ N .The rank function of M is defined as r M (a, b) := rank(M (a,b) ).Note that the multiplication of a matrix M ∈ C N ×N on the left with M corresponds to the downwards row operations and multiplication of M on the right with M + corresponds to the rightward column operations.Hence, one observes that M ∈ B πB + if and only if r M (a, b) = r π (a, b) for all (a, b) ∈ [N ] × [N ].The Zariski closure of the orbit X π := B πB + ⊆ C N ×N is called the matrix Schubert variety of π.Rothe presented a combinatorial technique for visualizing inversions of the permutation π.Definition 2.3.The Rothe diagram of π is defined as D(π) = {(π(j), i) : i < j, π(i) > π(j)}.One can draw the Rothe diagram D(π) in the following way: Consider the permutation matrix π in an N × N grid.Cross out each box containing 1 and all the other boxes to the south and east of each box containing 1.

( 1 )
G for the graph G is colored in green.The sets {1} and {3} are not in I(1)

Proposition 3 . 2 .Example 4 .
Y π is a T-variety of complexity d with respect to the torus action T if and only if dim(σ ∨ π ) = L (π) − d.Let us consider the matrix Schubert variety X π ∼ = Y π ×C 7 for π = [2143] in Example 1.The second figure represents L(π) and the third figure represents the bipartite graph G π .For each box (a, b) ∈ L(π), we construct an edge (a, b) ∈ E(G π ) with vertices a ∈ U 1 and b ∈ U 2 .The dimension of the associated dual edge cone σ ∨ π is 5 and |L (π)| = 7. Hence Y π is a T -variety of complexity 2 with respect to the effective torus action of T ∼ = (C * ) 5 and with a moment cone linearly equivalent to σ ∨ π .

Theorem 3 . 3 ([ 6 ,
Theorem 3.4]).Y π is a toric variety if and only if L (π) consists of disjoint hooks not sharing a row or a column.

Example 5 .
Let π = [2413] ∈ S 4 .The first figure illustrates the Rothe diagram D(π).The green colored boxes are L(π) and the purple colored boxes are L (π).The dimension of the associated bipartite graph and |L (π)| is three.Also, as seen in the last figure, L (π) has a hook shape.Thus, Y [2413] is a toric variety with respect to the effective torus action of T ∼ = (C * ) 3 , in particular the cone over the Segre variety P 1 × P 1 .
its alternative proof give us the opportunity to study the first-order deformations of Y π in terms of edge cone σ π and Rothe diagram D(π) in Section 4.

3 d 4 d 5 d 6 Figure 3 .
Figure 3. Compact 2-faces sharing an edge or a non-lattice vertex in Q(R)

Lemma 4 . 6 .
There exist k two-sided first independent sets of G π with | Ess(π)| = k + 1.Moreover, if k ≥ 2 and, C and C are two-sided first independent sets of G π , then C 1 C 1 and C 2 C 2 .

( 1 )
if and only if G[S] = A∈S G{A} has d + 1 connected components.Proposition 4.8.Let A = U 1 \{i}, B = U 2 \{j}, C = C 1 C 2 be three types of first independent sets of the bipartite graph G π .For any A, B ∈ I (1)

( 4 )( 2 )
Suppose that i ∈ C 1 and (A, C) / ∈ I G π .Consider the intersection subgraph G{A} ∩ G{C}.Similarly to last investigations, we conclude G[C 1 N (C 1 )] cannot admit {i} as its only isolated vertex.If C 1 = {i}, then the intersection subgraph admits of |N (C 1 )| + 1 isolated vertices and G[C 2 N (C 2 )].Assume that the intersection subgraph consists of the isolated vertex {i} C 1 and some vertex set C 2 N (C 1 ).This means that C := C 1 \{i} C 2 C 2 is a maximal two-sided independent set.Hence C ∈ I (1) G π .

Remark 4 . 4 . 3 .
In addition to the triple in Proposition 4.11, the triples of first independent sets of G π , containing the pairs in Proposition 4.8 (3) and (4) do not form a three-dimensional face of σ π .Classification of rigid toric varieties Y π .The following two results classify the rigid toric matrix Schubert varieties in terms of edge cone σ π and in terms of its Rothe diagram D(π).