Quadratic Gr\"obner bases of block diagonal matching field ideals and toric degenerations of Grassmannians

In the present paper, we prove that the toric ideals of certain $s$-block diagonal matching fields have quadratic Gr\"obner bases. Thus, in particular, those are quadratically generated. By using this result, we provide a new family of toric degenerations of Grassmannians.


INTRODUCTION
A toric degeneration of a given projective variety X is a flat family of varieties whose central fiber is a toric variety X 0 and all of whose general fibers are isomorphic to X . The resulting toric variety X 0 has a rich information on the original variety X . Hence, providing a toric degeneration is a useful tool to analyze algebraic varieties by using toric geometry. In the present paper, we study a new family of toric degeneration of Grassmannians.
Let K be a field. We use the notation Gr(r, n) for the Grassmannian, which is the space of r-dimensional subspaces of the n-dimensional vector space K n . The studies of toric degenerations of Grassmannians, flag varieties, Schubert varieties and Richardson varieties are an active research area in various branches of mathematics, such as algebraic geometry, algebraic topology, representation theory, commutative algebra, combinatorics, and so on. For the study of toric degenerations, see, e.g. [2,3,4,5,7,13,14] and so on. There are many ways to construct toric degenerations for them. The main example of toric degenerations of Gr(r, n) is, so-called, the Gelfand-Tsetlin degeneration (see [12,Section 14]). Moreover, the theory of Newton-Okounkov bodies can be applied to provide toric degenerations of varieties (see, e.g., [9] and the references therein). We can construct Newton-Okounkov bodies of Grassmannians from plabic graphs, which are certain bipartite graphs drawn in the disc, by a combinatorial manner ( [14]). Furthermore, tropical geometry is also used for the study of toric degenerations of defining ideals in general ( [11]). Namely, the top-dimensional cones of tropicalizations of varieties are good candidates to give toric degenerations, see [4,Lemma 1] and also [11]. It was also prove in [8] that the tropical cones arising from matching fields are a subfamily of Stiefel tropical linear spaces.
In addition, the theory of SAGBI bases can be also applied to provide toric degenerations. This is the tool we will use in the present paper. We refer the reader to [15,Section 11] for the introduction to the theory of SAGBI bases. For specifying the monomial order for SAGBI bases, we use matching fields in the sense of [16], which we will explain in Section 2. Note that the same approach of using matching fields to obtain toric degenerations of Schubert and Richardson varieties has been studied in [6] and [1], respectively.
Let us review the previous known results from [7,13] which are strongly related to our main theorems. In [13], Mohammadi and Shaw discuss a necessary condition for matching fields to provide a toric degeneration of Gr(r, n) as follows: [13,Theorems 1.2 and 1.3]). If the matching field Λ of Gr(r, n) produces its toric degeneration, then Λ is non-hexagonal. Moreover, the converse is also true for r = 3 under the assumption that the matching field ideal is quadratically generated.
(The terminologies will be defined in Section 2.) Hence, for the discussion of the existence of toric degenerations of Gr(3, n) from matching fields, the quadratic generation of the matching field ideals is quite important. It is mentioned in [13, Example 3.12 and Remark 3.13] that the matching field ideals are not necessarily quadratically generated. Thus, identifying families of matching fields with quadratic ideals is a natural problem. As a nice class of matching fields, s-block diagonal matching fields are introduced to be expected that their corresponding ideals are quadratically generated ([13, Definition 4.1]), and the following is proved: Taking those theorems into account, we prove the following main results of the present paper: Theorem 1.4 (See Theorems 4.2 and 5.3). Given a = (a 1 , . . . , a s ) ∈ Z s >0 with ∑ s i=1 a i = n and a i ∈ {1, 2} for i ∈ {1, s}, consider the s-block diagonal matching field Λ a . Then the matching field ideal J Λ a has a quadratic Gröbner basis. In particular, J Λ a is quadratically generated.
Moreover, the generating set {det(x I ) : I ∈ I r,n } of the Plücker algebra A r,n forms a SAGBI basis for A r,n with respect to the weight matrix M a associated with Λ a . Consequently, every Λ a gives rise to a toric degeneration of Gr(r, n).
We notice that Theorem 1.4 directly implies Theorem 1.3. Moreover, as we can see in Sections 4 and 5, the proofs become a little simpler than those of Theorem 1.3.
The paper is organized as follows: In Section 2, we prepare the necessary terminologies and the notation we will use. In Section 3, for the proof of our theorem, we recall the theory of toric ideals of the edge rings and their Gröbner bases. In Section 4, we will give a proof of the first half part of Theorem 1.4 (Theorem 4.2). In Section 5, we will give a proof of the second half part of Theorem 1.4 (Theorem 5.3). 2 Given integers r and n with 1 < r < n, let I r,n be the set of all r-subsets of [n] = {1, 2, . . ., n}. Let S = K[P I : I ∈ I r,n ] be the polynomial ring with n r variables. Let x = (x i j ) 1≤i≤r,1≤ j≤n be the r × n matrix of variables and let R = K[x] be the polynomial ring with rn variables. The Plücker ideal I r,n is defined by the kernel of the ring homomorphism ψ : S → R, P I → det(x I ), where x I denotes the r × r submatrix of x whose columns are indexed by I. The Plücker algebra A r,n is the image Im(ψ) of this map, which is isomorphic to S/I r,n . The Plücker algebra A r,n is well-known to be a homogeneous coordinate ring of the Plücker embedding of the Grassmannians Gr(r, n), called the Plücker embedding.
Let S r denote the symmetric group on [r]. An r × n matching field is a map Λ : I r,n → S r . For I = {i 1 , . . . , i r } ∈ I r,n with 1 ≤ i 1 < · · · < i r ≤ n and a matching field Λ, we associate the monomial of R where σ = Λ(I) ∈ S r . We define a ring homomorphism ψ Λ : S → R, ψ Λ (P I ) = sgn(Λ(I))x Λ(I) , where sgn(σ ) denotes the signature of σ ∈ S r . Then the matching field ideal J Λ of Λ is the kernel of ψ Λ . ). A matching field Λ is said to be coherent if there exists an r × n matrix M ∈ R r×n with its entries in R such that for every I ∈ I r,n the initial form in M (det(x I )) of det(x I ) with respect to M, which is the sum of all terms of det(x I ) having the lowest weights, is equal to ψ Λ (P I ). In this case, we call Λ a coherent matching field induced by M. For the matching field ideals, we use the notation J M instead of J Λ if Λ is a coherent matching field induced by M.
In the original definition [16, Section 1] of coherent matching fields, the initial form is set to be the sum of all terms having the highest weights, but we usually employ the definition with the sum of all terms having the lowest weights when it is related to the context of tropical geometry, following the convention of our main reference [13].
The main object of the present paper is the following matching fields: where α 0 = 0 and α k = ∑ k i=1 a i . Note that α s = n. Then the s-block diagonal matching field Λ a associated with a is defined by Λ a (I) = (1 2) if |I ∩ I q | = 1 where q = min{t : otherwise. 3 Any s-block diagonal matching field Λ a is coherent. In fact, Λ a is induced by the following weight matrix M a : Remark 2.4. (a) When a = (n), i.e., s = 1, the corresponding block diagonal matching field is so-called the diagonal matching field (see [16,Example 1.3]). This actually gives rise to the Gelfand-Tsetlin degeneration (see [12,Section 14]).
(b) In [7], the terminology "block diagonal" is used for "2-block diagonal" in the sense of Definition 2.2. Namely, the first example of Example 2.3 is not block diagonal in the sense of [7].
Let w a = (w 1 , . . . , w n ) be the second row of M a . Note that, for Thus w a is useful to study Λ a . Moreover w a satisfies In the present paper, we mainly consider s-block diagonal matching fields Λ a such that a i ∈ {1, 2} for all i / ∈ {1, s}. This is equivalent to the condition In particular, if s = 2, then the condition (2) is always satisfied. 4

EDGE RINGS OF BIPARTITE GRAPHS
In this section, we recall the notion of edge rings of bipartite graphs. We will see that the matching field ideal of a 2 × n matching field Λ is the toric ideal of a certain bipartite graph associated with Λ. We will also discuss Gröbner bases of matching field ideals. Consult [10, Chapter 5] and [18] for the introduction to the toric ideals of graphs and [10,15] for the introduction to the theory of Gröbner bases.
A graph G is said to be simple if G has no loops and no multiple edges An even cycle in the bipartite graph G of length 2q is a finite sequence of vertices of the form and there exist no repeated vertices. Given an even cycle C in (3), we write The following proposition is due to Villarreal [17, Proposition 3.1].
Proposition 3.1 ([10, Corollary 5.12]). Let G be a bipartite graph. Then the reduced Gröbner basis of I G with respect to any monomial order consists of the binomials of the form f C , where C is an even cycle in G. In particular, I G is generated by those binomials.
Let Λ M be the coherent matching field induced by a 2 × n weight matrix where w i = w j for any i = j, let G M be a bipartite graph on the vertex set {u 1 , . . ., u n } ⊔ {v 1 , . . ., v n } with the edge set It is easy to see that the matching field ideal J Λ M coincides with the toric ideal I G M of the bipartite graph G M . Let Then M diag is a weight matrix of a diagonal matching field. It is known [ is a (quadratic) Gröbner basis of J Λ with respect to <. In particular, J Λ is generated by quadratic binomials in G .
Let a = (a 1 , . . . , a s ) ∈ Z s >0 such that ∑ s i=1 a i = n, and let w a = (w 1 , . . ., w n ) be the second row of M a . Let G a = G M a . In order to prove our main theorem, we need a quadratic Gröbner basis of I G a with respect to a reverse lexicographic order defined as follows. Let < be a reverse lexicographic order on Then the reduced Gröbner basis of I G a with respect to the reverse lexicographic order < is Applying Buchberger's criterion [10, Theorem 1.29], it is enough to show that the S-polynomial S( f , g) of any two distinct binomials f and g in G reduces to 0 with respect to G . Note that the initial monomials of f and g are different. From [10, Lemma 1.27], if the initial monomials of f and g are relatively prime, then S( f , g) reduces to 0. Suppose that the initial monomials of f and g have exactly one common variable. Then the degree of S( f , g) is three. Suppose that the remainder h of S( f , g) with respect to G is not zero. By Proposition 3.1, we may assume that h is of the form is an even cycle of G of length 6. Let A = (a i j ) be the n × n matrix where Then the cycle C appears in A as one of the following submatrices of A: If B 1 = 1 0 0 1 is a submatrix of A corresponding to the k 1 , k 2 -th rows and the k 3 , k 4 -th columns of A, then we have w k 1 > w k 3 , w k 2 > w k 4 , w k 1 ≤ w k 4 and w k 2 ≤ w k 3 , a contradic- Thus it follows that, each matrix in (4) contains at most one 0, and hence C has at least two chords. If C has a chord at a position marked by * in then C has a submatrix of one of B = * 1 1 1 and 1 1 1 * . Then B corresponds to a cycle C ′ of length 4 in G and f C ′ belongs to G . Moreover, the initial monomial of f C ′ divides one of the monomials of h. This contradicts the hypothesis that h is a remainder with respect to G . Thus we may assume that where ℓ 1 < ℓ 2 < ℓ 3 and m 1 < m 2 < m 3 is a submatrix of A. Then Hence we have (1). This is a contradiction. Thus we have We now show that m 1 < ℓ 1 < ℓ 2 < m 3 < ℓ 3 . Suppose that (ℓ 2 <)ℓ 3 < m 3 . Since w ℓ 2 > w m 3 , we have w ℓ 2 > w ℓ 3 by condition (1), a contradiction. Suppose that ℓ 1 < m 3 < ℓ 2 . Since w ℓ 1 > w ℓ 2 , we have w m 3 > w ℓ 2 by condition (1), a contradiction. Suppose that (m 2 <) m 3 < ℓ 1 (< ℓ 2 ).

QUADRATIC GRÖBNER BASES OF s-BLOCK DIAGONAL MATCHING FIELDS
Recall that the matching field ideal J Λ of a matching field Λ is the kernel of a ring homomorphism defined by ψ Λ (P I ) = sgn(σ )x σ (1)i 1 · · · x σ (r)i r , where I = {i 1 , . . . , i r } ∈ I r,n with i 1 < · · · < i r and σ = Λ(I). From now on, we identify a variable P I with Lemma 4.1. Let Λ a be an s-block diagonal matching field of size r × n associated with a ∈ Z s >0 . Let Then we have the following: Proof. It is easy to see that, if a binomial f appearing in (i) -(vii) belongs to the polynomial ring S, then ψ Λ ( f ) = 0 and hence f ∈ J Λ a . Thus, it is enough to show that the second monomial of any binomial appearing in (i) -(vii) belongs to the polynomial ring S. First, we show that , then w j 1 > w j 2 , a contradiction. Hence Suppose that w i 1 ≤ w j 1 . It then follows that w i 1 ≤ w j 1 < w j 2 , a contradiction. Hence w i 1 > w j 1 .
Therefore, the second monomial of any binomial appearing in (i) -(vii) belongs to the polynomial ring S.
Let < be a reverse lexicographic order induced by the ordering of variables such that The following theorem is the first main result of the present paper: Let Λ a be an s-block diagonal matching field of size r × n associated with a = (a 1 , . . . , a s ) ∈ Z s >0 such that ∑ s i=1 a i = n and a i ∈ {1, 2} for i / ∈ {1, s}. Let G a be the set of all binomials appearing in Lemma 4.1 (i) -(vii). Then G a is a quadratic Gröbner basis of J Λ a with respect to the reverse lexicographic order <.
Proof. Suppose that G a is not a Gröbner basis of J Λ a . By [10,Theorem 3.11], there exists an irreducible homogeneous binomial u − v ∈ J Λ a such that neither u nor v belongs to the monomial ideal generated by the initial monomials of the binomials in G a . Note that the initial monomial of each binomial in G a is the first monomial in Lemma 4.1 (if the polynomial is not zero). Let If i µk > i ηk for some 3 ≤ k ≤ r and 1 ≤ µ < η ≤ d, then u is divisible by the initial monomial of a binomial appearing in Lemma 4.1 (i) -(vi). Thus we may assume that i 1k ≤ · · · ≤ i dk and j 1k ≤ · · · ≤ j dk 10 for all k = 2. Since u − v belongs to J Λ a , it follows that i µk = j µk for any 1 ≤ µ ≤ d and k = 2. If i µ2 = j µ2 for any 1 ≤ µ ≤ d, then u − v = 0, a contradiction. Hence is a nonzero binomial in I G a . We may assume that u is the initial monomial of u − v. From Proposition 3.3, there exists a binomial is a binomial belonging to G a whose initial monomial is the first monomial. Since the first monomial of f divides u, this is a contradiction.

TORIC DEGENERATIONS ASSOCIATED TO s-BLOCK DIAGONAL MATCHING FIELDS
In this section, we provide a new family of toric degenerations of Gr(r, n) (Corollary 5.6). For this goal, we recall the notion of SAGBI bases for the Plücker algebra.  The following is the second main result of the present paper and the essential statement for the existence of toric degenerations arising from s-block diagonal matching fields with certain conditions. Note that the case s = 2 of Theorem 5.3 was proved in [7,Theorem 4.3]. Moreover, as described below, the proof of Theorem 5.3 looks simpler than that of [7,Theorem 4.3].
Let Λ a be an s-block diagonal matching field associated with a = (a 1 , . . ., a s ) ∈ Z s >0 such that ∑ s i=1 a i = n. Let A a be the toric ring arising from Λ a , i.e., S/J Λ a , and let A 0 denote that of the diagonal matching field, i.e., A 0 = A (n) . Let [A] 2 denote the K-vector subspace of A generated by elements of degree 2 in A.
Our proof of Theorem 5.3 essentially consists of Theorem 4.2 and the following lemma. Note that the condition "a i ∈ {1, 2} for i / ∈ {1, s}" is not required in this lemma.
Proof. Let G a be the set of all binomials appearing in Lemma 4.1 (i) -(vii) and let < be the reverse lexicographic order defined by (6). Since we do not assume a i ∈ {1, 2} for i / ∈ {1, s}, G a is not necessarily a Gröbner basis with respect to <. Let in < (G a ) be the monomial ideal generated by the initial monomials of the binomials in G a . Then in < (G a ) is a subideal of the initial ideal in < (J Λ a ) of J Λ a . Let M be the set of all quadratic monomials in S \ in < (G a ) . It is known that the set of all quadratic monomials in S \ in < (J Λ a ) is a basis of [A a ] 2 (see, e.g., [15, Thus it is enough to show that (i) the cardinality of M is independent of a; (ii) M is linearly independent over K. First, we prove (i). Let Case 1: |{ℓ 1 , ℓ 2 , m 1 , m 2 }| = 2. Then ℓ 1 = m 1 and ℓ 2 = m 2 . Given positive integers 1 ≤ µ < ν ≤ n, the number of monomials u ∈ M of the form in ( * ) such that {ℓ 1 , ℓ 2 } = {µ, ν} is equal to |{(ℓ 3 , . . . , ℓ r , m 3 , . . . , m r ) : ℓ k < ℓ k+1 , m k < m k+1 , m k ≤ ℓ k for k ≥ 3, and ν < ℓ 3 }| since such a monomial is of the form ( * ), where ℓ k < ℓ k+1 , m k < m k+1 , m k ≤ ℓ k for k ≥ 3, and ν < ℓ 3 . The number of such monomials is independent of a. Case 2: |{ℓ 1 , ℓ 2 , m 1 , m 2 }| = 3.
Next, we prove (ii). On the contrary, suppose that M is linearly dependent over K.