Abstract
In this paper, we survey the theory of Cartwright–Sturmfels ideals. These are \(\mathbb {Z}^{n}\)graded ideals, whose multigraded generic initial ideal is radical. Cartwright–Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borelfixed point. This has consequences, e.g., on their universal Gröbner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright–Sturmfels ideals and we find a new one. Among determinantal ideals of samesize minors of a matrix of variables and Schubert determinantal ideals, we are able to characterize those that are Cartwright–Sturmfels.
Introduction
In 2010, Cartwright and Sturmfels published a paper [6] containing surprising results on certain multigraded ideals. More precisely, they proved that any \(\mathbb {Z}^{n}\)multigraded ideal that has the \(\mathbb {Z}^{n}\)multigraded Hilbert function of the ideal of 2minors of an m × n generic matrix must be radical and Cohen–Macaulay. During our stay at MSRI in 2012, we realised that a similar phenomenon was related to the universal Gröbner basis theorem for maximal minors proved in the nineties by Bernstein, Sturmfels, and Zelevinsky [3, 31]. We managed to identify a notion that “explains” the common features behind these two settings and that is flexible enough to be useful in other contexts. The key idea is to consider the family of multigraded ideals with radical multigraded generic ideals, that we named after Cartwright and Sturmfels. We wrote four papers related to the subject [8,9,10,11]. The goal of this note is to give a short introduction to Cartwright–Sturmfels ideals, to highlight their properties, and to present some classes of Cartwright–Sturmfels ideals, both old and new. In particular, in Section 4 we classify determinantal ideals that are Cartwright–Sturmfels in the generic case and derive results for the nongeneric case. In Section 5 we characterize Schubert determinantal ideals that are Cartwright–Sturmfels. In Section 6 we take the occasion to correct a mistake in the proof of Theorem 2.1 of [9] asserting that any binomial edge ideals is Cartwright–Sturmfels. Finally, in Section 7 we recall another result from [9] asserting that the multiprojective closure of any linear ideal is Cartwright–Sturmfels and conclude with a question suggested by it.
The authors thank Anna Weigandt and Patricia Klein for useful discussions on the material of this paper.
Multigraded Generic Initial Ideals and Multidegree
Let \(n\in \mathbb {N}_{+}\) and \(m_{1},\ldots ,m_{n}\in \mathbb {N}\). Let S = K[x_{ij}  1 ≤ j ≤ n,0 ≤ i ≤ m_{j}] be a polynomial ring over a field K endowed with the standard \(\mathbb {Z}^{n}\)grading induced by setting \(\deg (x_{ij})=e_{j}\), where \(e_{j}\in \mathbb {Z}^{n}\) is the jth standard basis vector.
We will deal with \(\mathbb {Z}^{n}\)graded ideals and modules of S. We use the words \(\mathbb {Z}^{n}\)graded and multigraded interchangeably. For simplicity we always assume the term orders on S satisfy \(x_{0j}>x_{1j}>\cdots >x_{m_{j}j}\) for all j = 1,…,n.
The ring S may be thought of as the coordinate ring of the product of n projective spaces, i.e.
A multigraded prime ideal P of S is relevant if P does not contain \(S_{(1,1,\dots ,1)}\) and irrelevant otherwise. When K is algebraically closed relevant prime ideals correspond to irreducible subvarieties of \(\mathbb {P}^{(m_{1},\dots ,m_{n})}\).
The Multigin
The group \(G=\text {GL}_{m_{1}+1}(K)\times \cdots \times \text {GL}_{m_{n}+1}(K)\) acts naturally on S as the group of multigraded Kalgebra automorphisms, i.e., coordinate changes that fix each factor in the product of projective spaces. Let I be a multigraded ideal of S and let σ be a term order on S. As in the standard \(\mathbb {Z}\)graded situation, if K is infinite there exists a nonempty Zariski open \(U\subseteq G\) such that \(\text {in}_{\sigma }(gI)=\text {in}_{\sigma }(g^{\prime } I)\) for all \(g,g^{\prime }\in U\). This leads to the definition of multigraded generic initial ideal. We refer the reader to [13, Theorem 15.23] for details on the generic initial ideals in the \(\mathbb {Z}\)graded case and to [1, Section 1] for a similar discussion in the \(\mathbb {Z}^{n}\)graded case.
Definition 2.1
The multigraded generic initial ideal gin_{σ}(I) of I with respect to σ is the ideal in_{σ}(gI), where g is a generic multigraded coordinate change, i.e. g ∈ U and U is a nonempty Zariski open subset of G.
Let \(B=B_{m_{1}+1}(K)\times {\cdots } \times B_{m_{n}+1}(K)\) be the Borel subgroup of G, consisting of the upper triangular invertible matrices in G. One knows that gin_{σ}(I) is Borel fixed, that is, it is fixed by the action of every g ∈ B.
Multidegree and Dual Multidegree
For a finitely generated \(\mathbb {Z}^{n}\)graded module \(M=\oplus _{a\in \mathbb {Z}^{n}} M_{a}\) over a standard \(\mathbb {Z}^{n}\)graded polynomial ring S, one may define the multigraded Hilbert function as the function HF(M,−) that associates to \(a=(a_{1},\dots , a_{n}) \in \mathbb {Z}^{n}\) the number \(\text {HF}(M,a)=\dim _{K} M_{a}\). As in the \(\mathbb {Z}\)graded case, for a ≫ 0 the multigraded Hilbert function agrees with a polynomial in n variables \(P_{M}(Z)=P_{M}(Z_{1},\dots , Z_{n})\), the multigraded Hilbert polynomial of M.
Let d(M) be the total degree of P_{M}(Z). Under mild assumptions, for example when all the minimal primes of M are relevant, one has that \(d(M)=\dim (M)n\). The homogeneous component of degree d(M) of P_{M}(Z) can be written as
where the sum ranges over all \(b\in \mathbb {N}^{n}\) with b_{i} ≤ m_{i} such that b = d(M). The numbers e_{b}(M) are the multidegrees (or mixed multiplicities) of M. It turns out that they are nonnegative integers. The multidegree of M is the polynomial
where the sum is over all \(b\in \mathbb {N}^{n}\) such that b = d(M).
One can regard M as a \(\mathbb {Z}\)graded module by M_{v} = ⊕_{a=v}M_{a}. With respect to this \(\mathbb {Z}\)grading, M has an ordinary multiplicity e(M) and, if all the minimal primes of M are relevant, one has
where the sum ranges over all the \(b\in \mathbb {N}^{n}\) such that \(b=\dim (M)n\). This is proved in [7, Theorem 2.8], but a special case appears already in [32].
When M is the coordinate ring of an irreducible multiprojective variety \(X \subseteq \mathbb {P}^{(m_{1},\dots ,m_{n})}\) over an algebraically closed field, the multidegrees e_{b}(M)’s have a geometric interpretation. Indeed, in that case e_{b}(M) is the number of points of \(\mathbb {P}^{(m_{1},\dots ,m_{n})}\) that one gets by intersecting X with \(L_{1}\times L_{2} \times {\dots } \times L_{n}\) where each L_{i} is a general linear subspace of \(\mathbb {P}^{m_{i}}\) of codimension b_{i}.
Definition 2.2
A \(\mathbb {Z}^{n}\)graded module M has a multiplicityfree multidegree if e_{b}(M) ∈{0,1} for all b with b = d(M).
The relevant prime ideals P of S such that S/P have a multiplicityfree multidegree are studied in [4] by Brion (in a more general setting) who proves in particular that S/P is Cohen–Macaulay.
The word multidegree is used in the literature also to refer to another polynomial invariant of M, which we call dual multidegree throughout this paper, in order to avoid confusion. The dual multidegree is defined as follows: The multigraded Hilbert series of M is
Let
It turns out that \(K_{M}(Z) \in \mathbb {Z}[Z_{1}^{\pm 1},\dots ,Z_{n}^{\pm 1}]\). The dual multidegree \(\text {Deg}^{\ast }_{M}(Z)\) of M is the homogeneous component of smallest total degree of \(K_{M}(1Z_{1},\dots ,1Z_{n})\). One can show that \(\text {Deg}^{\ast }_{M}(Z)\in \mathbb {N}[Z_{1},\dots ,Z_{n}]\). Notice that the dual multidegree corresponds to the multidegree as defined in e.g. [24, 27].
Multidegrees of Radical Monomial Ideals
Let J be a radical monomial ideal of S with associated simplicial complex \({\varDelta } \subseteq 2^{T}\). Here T = {(i, j) : 1 ≤ j ≤ n,0 ≤ i ≤ m_{j}}. The ideal J is naturally \(\mathbb {Z}^{T}\)graded, however here we consider its \(\mathbb {Z}^{n}\)graded structure and describe its multigraded Hilbert polynomial in terms of Δ. For each F ∈Δ and j ∈ [n] we set
and \(c(F)=(c_{1}(F),\dots , c_{n}(F))\in \mathbb {N}^{n}\). A face F is relevant if the corresponding prime ideal (x_{ij} : (i, j)∉F) is relevant, i.e., c_{j}(F) > 0 for every \(j=1,\dots , n\). Let us denote by R(Δ) the set of the relevant faces of Δ, i.e.,
Lemma 2.3
For every \(a=(a_{1},\dots , a_{n})\in \mathbb {N}_{+}^{n}\) one has
in particular
Proof
First we observe that HF(S/J, a) is the number of monomials in S of multidegree a which are not contained in J. To a monomial \(x^{v}=\prod x_{ij}^{v_{ij}}\) we may associate its support F(x^{v}) = {(i, j) : v_{ij} > 0}. Since a_{i} > 0 for all i, by construction we have x^{v}∉J and \(\deg (x^{v})=a\) if and only if F(x^{v}) ∈ R(Δ). We partition the set of monomials of degree a which do not belong J according to their support. The monomials of degree a supported on a given F ∈ R(Δ) have the form \(({\prod }_{(i,j)\in F} x_{ij} )x^{v}\), where x^{v} is a monomial with support contained in F and degree a − c(F). The number of these monomials is \({\prod }_{j=1}^{n} \left (\begin {array}{c}{a_{j}1}\\{c_{j}(F)1} \end {array}\right )\). □
Denote by \(\mathbb {F}({\varDelta })\) the set of the facets of Δ. Recall that Δ is a pure simplicial complex if all the facets of Δ have the same dimension. As an immediate corollary we have
Lemma 2.4
Assume that Δ is a pure simplicial complex and that \(\mathbb {F}({\varDelta }) \cap R({\varDelta }) \neq \emptyset \). Then
Cartwright–Sturmfels Ideals
In this section we recall the definition of Cartwright–Sturmfels ideals and some facts about them, which were discussed in our papers [8,9,10,11].
Definition 3.1
A multigraded ideal I of S is a Cartwright–Sturmfels ideal if there exists a radical Borelfixed multigraded ideal which has the same multigraded Hilbert series as I. If the ground field is infinite, this is equivalent to the fact that I has a radical multigraded generic initial ideal [11, Proposition 2.6].
We denote by CS(S), or simply by CS if S is clear from the context, the family of Cartwright–Sturmfels ideals of S.
Example 3.2
The \(\mathbb {Z}\)graded Cartwright–Sturmfels ideals are exactly those generated by linear forms. In fact, if I is not generated by linear forms, let d > 1 be the least degree of a minimal generator of I which is not linear. Then the generic initial ideal of I has a minimal generator which is the dth power of a variable. In particular, the generic initial ideal of I with respect to any term order is not radical.
Notice that the property of being Cartwright–Sturmfels depends on the multigrading.
Example 3.3
If \(I\subseteq S\) is generated by a nonzero element of degree \((1,1,\dots , 1)\in \mathbb {Z}^{n}\) then I is a \(\mathbb {Z}^{n}\)graded Cartwright–Sturmfels ideal for obvious reasons. However, as we have observed in Example 3.2, the ideal I is not a \(\mathbb {Z}\)graded Cartwright–Sturmfels ideal if n > 1.
Cartwright–Sturmfels ideals have many interesting properties. The next proposition summarizes some of them.
Proposition 3.4
Let I ∈CS and let J be a radical Borel fixed ideal such that HF(I, a) = HF(J, a) for all \(a\in \mathbb {N}^{n}\). Then:

(1)
I is radical and gin_{τ}(I) = J for every term order τ [11, Proposition 2.6].

(2)
in_{τ}(I) ∈CS, in particular it is square free, for every term order τ [11, Remark 2.5].

(3)
reg(I) ≤ n [11, Corollary 2.15].

(4)
If K is algebraically closed, then P ∈CS for every minimal prime P of I [4], see also [10, Corollary 1.12].

(5)
I is generated by elements of multidegree ≤ (1,…,1) [11, Proposition 2.6].

(6)
All reduced Gröbner bases of I consist of elements of multidegree ≤ (1,…,1). In particular, I has a universal Gröbner basis of elements of multidegree ≤ (1,…,1) [11, Proposition 2.6].
The family CS is closed under some natural operations.
Proposition 3.5 (11, Theorem 2.16)
Let L be a \(\mathbb {Z}^{n}\)homogeneous linear form of S. In the following S/(L) is identified with a polynomial ring with the induced \(\mathbb {Z}^{n}\)graded structure. Let \(U_{i}\subseteq S_{e_{i}}\) be vector subspaces for all i = 1,…,n and let R = K[U_{1},…,U_{n}] be the \(\mathbb {Z}^{n}\)graded polynomial subring of S that they generate. Then:

(1)
If I ∈CS(S), then I : L ∈CS(S).

(2)
If I ∈CS(S), then I + (L) ∈CS(S) and I + (L)/(L) ∈CS(S/(L)).

(3)
If I ∈CS(S), then I ∩ R ∈CS(R).
Moreover one has
Proposition 3.6

(1)
If I ∈CS(S) then S/I has multiplicityfree multidegree.

(2)
Vice versa, suppose that K is algebraically closed, I is a relevant prime ideal of S, and S/I has multiplicityfree multidegree. Then I ∈CS(S).
Part (2) is proved in [4] using a different terminology.
Determinantal Cartwright–Sturmfels Ideals
The goal of this section is to discuss Cartwright–Sturmfels ideals that are generated by minors of matrices. New results on the family of Schubert determinantal ideals will be presented in Section 5.
We start by discussing generic determinantal ideals, i.e., ideals of samesize minors of the matrix of variables. Let X = (x_{ij}) be an m × n matrix of variables over a field K and S = K[x_{ij} : 1 ≤ j ≤ n and 1 ≤ i ≤ m]. We consider the \(\mathbb {Z}^{n}\)graded structure on S induced by \(\deg (x_{ij})=e_{j}\in \mathbb {Z}^{n}\). In the notation of Section 2 we have m_{j} = m − 1 for \(j=1,\dots , n\) and, in accordance with usual notation for matrices, the index i varies from 1 to m. Let I_{t}(X) be the ideal of S generated by the tminors of X. Clearly I_{t}(X) is \(\mathbb {Z}^{n}\)graded and our first goal is to compute the multidegree of S/I_{t}(X).
The multigraded Hilbert function, hence the multidegree, does not change if we replace I_{t}(X) with an initial ideal. The ideal I_{t}(X) has a wellknown square free initial ideal, discussed in [5, 21, 30]. It is the ideal generated by the products of the entries on the main diagonals of the tminors, whose associated simplicial complex will be denoted by π_{t}. The facets of π_{t} can be identified with the families of nonintersecting paths in the grid [m] × [n] from the starting points p_{1} = (1,n),p_{2} = (2,n),…,p_{t− 1} = (t − 1,n) to the endpoints q_{1} = (m,1),q_{2} = (m,2),…,q_{t− 1} = (m, t − 1).
For example, for m = 4, n = 5, and t = 3, we have p_{1} = (1,5), p_{2} = (2,5), q_{1} = (4,1), and q_{2} = (4,2). The following is a facet of π_{3}:
depicted using the matrix coordinates and marking with “1” the lattice points which belong to the first path (from p_{1} to q_{1}) and with “2” the lattice points of the second path (from p_{2} to q_{2}).
Each family of nonintersecting paths must have at least t − 1 points on each column. Therefore, each facet of π_{t} is relevant if t > 1. With the notation of Section 2, \(\mathbb {F}({\Pi }_{t})\subseteq R({\Pi }_{t})\). Summing up, by Lemma 2.4 we have
where c_{j}(F) = {(a, b) ∈ F : b = j}.
We introduce the generating function associated to the statistics c(F). Given a collection U of subsets of [m] × [n] we set
so that we may rewrite (4.1) as
Next, we give a determinantal formula for \(W(\mathbb {F}({\Pi }_{t}),Z_{1},\dots , Z_{n})\). One observes that the Gessel–Viennot involution [17], used in [21] to compute \(\mathbb {F}({\Pi }_{t})\), is compatible with any weight given to the lattice points. Hence one gets immediately
where Paths(p_{i},q_{j}) is the set of the paths from p_{i} to q_{j}.
In the sequel, h_{v}(L) denotes the complete homogeneous symmetric polynomial of degree v on the set L, i.e., the sum of all monomials of degree v in the elements of L.
Lemma 4.1
Given p = (a, b) and q = (c, d) with a ≤ c and b ≥ d, we have
Proof
Any path P from p to q is uniquely determined by the the number of points of intersection with the columns. Any such path must have at least one point on column j for all d ≤ j ≤ b, and no points on the other columns. The only other constraint is that the path has (b − d) + (c − a) + 1 points in total. In terms of \(c(P)=(c_{1}(P),\dots , c_{n}(P))\) the constraints are c_{j}(P) > 0 if and only if d ≤ j ≤ b and \({\sum }_{j=d}^{b} c_{j}(P)=(bd)+(ca)+1\). Expressing this in terms of generating functions yields the desired result. □
We can now compute the multidegree of generic determinantal rings.
Theorem 4.2
The multidegree of the determinantal ring S/I_{t}(X) of an m × n matrix of variables X with \(2\leq t\leq \min \limits \{m,n\}\) and with respect to the \(\mathbb {Z}^{n}\)graded structure induced by \(\deg (x_{ij})=e_{j}\) is
Proof
Combining (4.2) and (4.3) with Lemma 4.1 we obtain
For \(j=1,\dots ,t1\), the factor \({\prod }_{k=j}^{n} Z_{k} \) can be extracted from the determinant, hence the monomial in front of the determinant becomes
while the determinant becomes
It now suffices to prove that the latter equals
Let us explain this last equality in full detail in the case t = 4, which is general enough to show all the relevant features. We wish to prove the equality
In the first term of (4.4), we subtract the second column from the third and the first column from the second. Since \(h_{v}(Z_{j+1},\dots , Z_{n})h_{v}(Z_{j},\dots , Z_{n})=Z_{j} h_{v1}(Z_{j},\dots , Z_{n})\), we can factor out − Z_{2} from the third column and − Z_{1} from the second. The first term of (4.4) therefore becomes
Subtracting the second column from the third and factoring − Z_{1}, we obtain
Finally we exchange rows one and three and then transpose. This yields
which is the second term of (4.4). □
The determinant that appears in the statement of Theorem 4.2 is a Schur polynomial. We refer to [26] and [29] for a treatment of the theory of symmetric functions and Schur polynomials. Here we collect only the definitions and the properties that we will use in the sequel. A partition λ is a weakly decreasing sequence of nonnegative integers λ_{1},λ_{2},…,λ_{r}. Given a partition λ = λ_{1},λ_{2},…,λ_{r} the Schur polynomial s_{λ}(Z) associated λ and with respect to the variables
is
By construction s_{λ}(Z) is a symmetric homogeneous polynomial of degree \(\lambda ={\sum }_{i=1}^{r} \lambda _{i}\) with integral coefficients. Denote by m_{μ}(Z) the monomial symmetric polynomial associated with the partition \(\mu =\mu _{1}\geq \mu _{2} \geq {\dots } \geq \mu _{n}\geq 0\), that is, the sum of the monomials in the \(\mathcal {S}_{n}\)orbit of \(Z_{1}^{\mu _{1}}{\cdots } Z_{n}^{\mu _{n}}\).
Since the m_{μ}(Z)’s form a Kbasis of the space of symmetric polynomials, one can express s_{λ}(Z) as
The coefficients K_{λ, μ} are known as the Kostka numbers of the pair of partitions λ, μ. We recall their main properties. We refer to [29, p. 309] for the definition of semistandard (Young) tableau.
Proposition 4.3
For every pair of partitions \(\lambda =\lambda _{1},\lambda _{2}, \dots , \lambda _{r}\) and \(\mu =\mu _{1}, \mu _{2}, \dots , \mu _{n}\) with λ = μ one has:

(1)
\(K_{\lambda , \mu }\in \mathbb {N}\).

(2)
K_{λ, μ} > 0 if and only if λ ≥ μ in the dominance order, i.e., \({\sum }_{i=1}^{s} \lambda _{i} \geq {\sum }_{i=1}^{s} \mu _{i}\) for every \(s=1,\dots , r\).

(3)
K_{λ, μ} is the number of semistandard tableaux of shape λ with entries \(1,\dots ,n\) and multiplicities given by μ (i.e., μ_{1} entries are equal to 1, μ_{2} entries are equal to 2, and so on).
This allows us to reformulate Theorem 4.2 in terms of Schur polynomials.
Theorem 4.4
The multidegree of the determinantal ring S/I_{t}(X) of an m × n matrix of variables X with \(2\leq t\leq \min \limits \{m,n\}\) and with respect to the \(\mathbb {Z}^{n}\)graded structure induced by \(\deg (x_{ij})=e_{j}\) is
where
Summing up, we have the following combinatorial description of the multidegrees of determinantal rings.
Theorem 4.5
Let X be an m × n matrix of variables and consider S = K[X] with the \(\mathbb {Z}^{n}\)graded structure induced by \(\deg (x_{ij})=e_{j}\). Let \(2\leq t\leq \min \limits \{m,n\}\) and let S/I_{t}(X) be the associated \(\mathbb {Z}^{n}\)graded determinantal ring. Set λ = ℓ^{(t− 1)} with ℓ = m + 1 − t. For \(b\in \mathbb {N}^{n}\) with \(b=\dim S/I_{t}(X)n\), the multidegrees e_{b}(S/I_{t}(X)) satisfy the following properties:

(1)
e_{b}(S/I_{t}(X)) is a symmetric function of b.

(2)
e_{b}(S/I_{t}(X)) > 0 if and only if t − 2 ≤ b_{i} ≤ m − 1 for every \(i=1,\dots , n\).

(3)
Set c_{i} = b_{i} + 1 for \(i=1,\dots , n\). Then e_{b}(S/I_{t}(X)) is the number of families of nonintersecting paths from \(p_{1}=(1,n), p_{2}=(2,n), \dots , p_{t1}=(t1,n)\) to \(q_{1}=(m,1), q_{2}=(m,2), {\dots } , q_{t1}=(m, t1)\) with exactly c_{i} points on the ith column for \(i=1,\dots , n\).

(4)
Set μ_{i} = b_{i} − (t − 2) for \(i=1,\dots , n\). Then e_{b}(S/I_{t}(X)) equals the Kostka number K_{λ, μ}, that is, the number of semistandard tableaux of shape λ with entries \(1,\dots ,n\) and multiplicities given by μ.
Proof
(1) The fact that e_{b}(S/I_{t}(X)) is a symmetric function of b follows from the fact that I_{t}(X) is invariant under the permutation of the columns or from Theorem 4.2. Furthermore, (3) and (4) are reformulations of Theorem 4.2 and (4.2). Finally (2) follows by applying Proposition 4.3 part (2). □
Example 4.6
For m = n = 4 and t = 3 we have that ℓ = m + 1 − t = 2, t − 1 = 2 and λ = 2,2. With \(Z=Z_{1},\dots , Z_{4}\) and \(z={\prod }_{i=1}^{4} Z_{i}\) we have
By Proposition 4.3, the coefficients appearing in the expression have two combinatorial interpretations. For example, the coefficient 2 of m_{(2,2,2,2)} is the Kostka number K_{22,1111}, i.e., the number of semistandard tableaux of shape 2,2 with entries \(1,\dots ,4\) and multiplicities given by (1,1,1,1):
Moreover, it is also the number of families of nonintersecting paths from p_{1} = (1,4), p_{2} = (2,4) to q_{1} = (4,1), q_{2} = (4,2) with 3 points on each column:
We have also a geometric interpretation of the coefficient of m_{(2,2,2,2)}: it is the number of points that one gets by intersecting the variety of 4 × 4 matrices of rank at most 2, regarded as a multigraded subvariety of \((\mathbb {P}^{3})^{4}\), with \(L_{1}\times {\dots } \times L_{4}\). Here each L_{i} is a generic linear space of codimension 2 of \(\mathbb {P}^{3}\). Interpreting the four columns of the matrix as points in \(\mathbb {P}^{3}\), the rank 2 condition means that the four points belong to a line in \(\mathbb {P}^{3}\) that must intersect the four generic lines \(L_{1},\dots , L_{4}\). How many lines intersect four general lines in \(\mathbb {P}^{3}\)? The answer is 2 and this a classical instance of Schubert calculus, see [14, Sect. 3.4.1] for a modern exposition.
The computation in Example 4.6 appears also in [32] and it can be easily generalised.
Example 4.7
For m = n and t = n − 1 we have that ℓ = m + 1 − t = 2 and λ = 2^{(n− 2)}. With \(Z=Z_{1},\dots , Z_{n}\) and \(z={\prod }_{i=1}^{n} Z_{i}\) we have
with
Remark 4.8
In [27, Chapter 15] the authors compute the multidegree for a large family of determinantal ideals, the Schubert determinantal ideals, with respect to the finer multigrading \(\deg X_{ij}=(e_{i},f_{j}) \in \mathbb {Z}^{m}\oplus \mathbb {Z}^{n}\) with \(\{e_{1},\dots ,e_{m}\}\) and \(\{f_{1},\dots ,f_{n}\}\) being the canonical bases of \(\mathbb {Z}^{m}\) and \(\mathbb {Z}^{n}\). The ideal I_{t}(X) is a Schubert determinantal ideals. For them the authors observe in that the multidegree is given by a Schur polynomial [27, 15.39] (also known as supersymmetric Schur polynomial, see [25]) which is, at least apparently, different from the Schur polynomial that we have identified.
We are ready to state the main consequence of Theorem 4.5.
Theorem 4.9
Let S/I_{t}(X) be the determinantal ring of the m × n matrix of variables X with the \(\mathbb {Z}^{n}\)graded structure induced by \(\deg (x_{ij})=e_{j}\), with \(2\leq t\leq \min \limits \{m,n\}\). Then S/I_{t}(X) has a multiplicityfree multidegree if and only if t = 2 or \( t=\min \limits \{m,n\}\).
Proof
We need to prove two assertions:
Claim 1. If t = 2 or \( t=\min \limits \{m,n\}\), then e_{b}(S/I_{t}(X)) ∈{0,1} for all \(b\in \mathbb {Z}^{n}\) with \(b=\dim S/I_{t}(X)n\).
Claim 2. If \(2<t<\min \limits \{m,n\}\), then there exists \(b\in \mathbb {Z}^{n}\) with \(b=\dim S/I_{t}(X)n\) such that e_{b}(S/I_{t}(X)) > 1.
Claim 1 for t = 2 follows immediately from Theorem 4.2. Claim 1 for t = m ≤ n follows from the description of e_{b}(S/I_{t}(X)) in Theorem 4.5(4) in terms of semistandard tableau, since the corresponding shape is a single column. Finally, Claim 1 for t = n ≤ m can be treated as follows. By Theorem 4.5(1), e_{b}(S/I_{n}(X)) > 0 if and only if n − 2 ≤ b_{i} ≤ m − 1 and \(b=\dim S/I_{n}(X)n\). Setting b_{i} = (m − 1) − c_{i} and rewriting the conditions with respect to \((c_{1},\dots , c_{n})\), we have that e_{b}(S/I_{n}(X)) > 0 if and only if \((c_{1},\dots , c_{n})\in \mathbb {N}^{n}\) and \({\sum }_{i=1}^{n} c_{i}=mn+1\). Hence, there are exactly \(\left (\begin {array}{c}{m}\\{n1} \end {array}\right )\) elements \(b\in \mathbb {N}^{n}\) such that e_{b}(S/I_{n}(X)) > 0. By (2.1), \({\sum }_{b} e_{b}(S/I_{n}(X))\) gives the ordinary multiplicity of S/I_{n}(X), which is \(\left (\begin {array}{c}{m}\\{n1} \end {array}\right )\). It follows that e_{b}(S/I_{n}(X)) = 1 whenever e_{b}(S/I_{n}(X)) > 0.
For Claim 2, by Theorem 4.5(4) it suffices to show that there exists a \(\mu \in \mathbb {N}^{n}\) with μ = (t − 1)ℓ, ℓ = m + 1 − t, and such that there are at least two semistandard tableaux of shape ℓ^{(t− 1)} and entries \(1,\dots , n\) with multiplicities given by μ. One can take
For \(i=1,\dots , t3\), the ith row of any semistandard tableau of shape ℓ^{(t− 1)} and multiplicities given by μ consists of exactly ℓ entries equal to i. So we may simply assume that t = 3. Similarly, we may assume that n = t + 1 = 4, so that μ = (ℓ − 1,ℓ − 1,1,1). Now for \(j=1,\dots , m4\) the jth column must have entries 1 and 2. Again we may then assume that m = 4, hence ℓ = 2. Now it is clear that there are exactly two tableaux of shape 2,2 and multiplicities given by (1,1,1,1), namely those described in Example 4.6. □
As a corollary of Theorem 4.9 we have
Corollary 4.10
Let S/I_{t}(X) be the determinantal ring of the m × n matrix of variables X with the \(\mathbb {Z}^{n}\)graded structure induced by \(\deg (x_{ij})=e_{j}\), with \(1\leq t\leq \min \limits \{m,n\}\). Then I_{t}(X) is Cartwright–Sturmfels if and only if t = 1,2 or \(t=\min \limits \{m,n\}\).
Proof
The case t = 1 is obvious, so we may assume t > 1. Since I_{t}(X) is a relevant prime, the conclusion follows combining Theorem 4.9 and Proposition 3.6. □
The fact that I_{2}(X) is Cartwright–Sturmfels has been proved directly (i.e., without using Proposition 3.6) by Cartwright and Sturmfels in [6], hence the name. For I_{t}(X) with \(t=\min \limits \{m,n\}\) it has been proved directly in [8].
Combining Corollary 4.10 with Proposition 3.5, which states that any multigraded linear section of a Cartwright–Sturmfels ideal remains Cartwright–Sturmfels, we obtain the following result, originally proved in [11, Main Theorem].
Theorem 4.11
Let A = (a_{ij}) be an m × n matrix whose entries are \(\mathbb {Z}^{n}\)multigraded with \(\deg (a_{ij})=e_{j}\) for all i, j. Let I_{t}(A) be the ideal of tminors of A. Then I_{t}(A) is Cartwright–Sturmfels for \(t=1,2,\min \limits \{m,n\}\).
In particular when \(t=1,2,\min \limits \{m,n\}\) then I_{t}(A) has all the properties of the Cartwright–Sturmfels ideals listed in Proposition 3.4. When m ≤ n every maximal minor of A has a different \(\mathbb {Z}^{n}\)degree and we obtain a more precise statement.
Corollary 4.12
Under the assumptions of Theorem 4.11, if m ≤ n then the maximal minors of A form a universal Gröbner basis of I_{m}(A).
Remark 4.13
It is natural to ask whether the ideal I_{t}(A) can be Cartwright–Sturmfels, under the assumptions of Theorem 4.11 and for \(2<t<\min \limits \{m,n\}\). The answer is yes if A is very special (for example when I_{t}(A) = 0) and no for a general enough A (for example if I_{t}(A) has the expected codimension). Nevertheless, notice that the generators of I_{t}(A) have squarefree \(\mathbb {Z}^{n}\)degrees, hence cannot have factors of multiplicity larger than one. This suggest that I_{t}(A) might always be radical. It turns out that this is not the case: In [12, Example 7.2] the authors give examples of nonradical coordinate sections of determinantal ideals.
The multidegree of S/I_{t}(A) for \(t=2,\min \limits \{m,n\}\) was essentially computed in [9]. Indeed, in that paper we computed the prime decomposition of the multigraded gin of I_{t}(A), from which the multidegree is easily derived.
Schubert Determinantal Ideals and Matrix Schubert Varieties
Matrix Schubert varieties were introduced by Fulton in [16]. They are defined by rank conditions. In this section, we show that many defining ideals of matrix Schubert varieties are Cartwright–Sturmfels. We start by fixing the notation and recalling the definitions.
Let X = (x_{i, j}) be an n × n matrix of variables over a field K and let S = K[X] = K[x_{ij} : 1 ≤ i, j ≤ n]. We consider the \(\mathbb {Z}^{n}\)graded structure on S induced by \(\deg (x_{ij})=e_{j}\in \mathbb {Z}^{n}\). In the notation of Section 2, this corresponds to letting m_{j} = n − 1 for \(j=1,\dots , n\). For M a matrix of size n × n and a, b ∈{1,…,n}, let M_{a×b} be the submatrix of M consisting of the entries in position (i, j) where i ≤ a and j ≤ b.
Denote by \(\mathcal {S}_{n}\) the group of permutations on the set {1,…,n} and let \(\omega \in \mathcal {S}_{n}\). We write ω in line notation, i.e., ω = ω_{1}⋯ω_{n} if ω(i) = ω_{i}. We associate to ω the rank function \(r_{\omega }:\{1,\ldots ,n\}^{2}\to \mathbb {N}\) defined by
where (k, ℓ) ≤ (i, j) is the coefficentwise inequality. In other words, let P_{ω} be the permutation matrix corresponding to ω, that is, \(P_{\omega } e_{j}=e_{\omega _{j}}\). Then r_{ω}(i, j) is the number of ones in the submatrix (M_{ω})_{i×j}. Notice that this is the transpose of the usual definition of rank function, see e.g. [24, Section 1.3]. We choose this notation in order to be coherent with the \(\mathbb {Z}^{n}\)grading that we defined in Section 2.
Definition 5.1
Let \(m{\kern .5pt} \leq {\kern .5pt} n\) and let \(\omega {\kern .5pt} ={\kern .5pt} \omega _{1}\cdots \omega _{n}{\kern .5pt} \in {\kern .5pt} \mathcal {S}_{n}\) and \(\upsilon {\kern .5pt} ={\kern .5pt} \upsilon _{1}\cdots \upsilon _{m}\in \mathcal {S}_{m}\). We say that ω contains υ if there are 1 ≤ i_{1} < … < i_{m} ≤ n such that \(\omega _{i_{j}}<\omega _{i_{\ell }}\) if and only if υ_{j} < υ_{ℓ}.
Else, we say that ω avoids υ. A permutation \(\omega \in \mathcal {S}_{n} \) is vexillary if it avoids the permutation \(2143\in \mathcal {S}_{4}\). In particular every permutation in \(\mathcal {S}_{n}\) with n ≤ 3 is vexillary.
To each permutation, one associates a Rothe diagram and an essential set as follows.
Definition 5.2
The Rothe diagram associated to \(\omega \in \mathcal {S}_{n}\) is
The essential set of ω is
Notice that, as for the rank function, these are the transpose of the usual Rothe diagram and essential set of a permutation.
Example 5.3
Let \(\omega =1432\in \mathcal {S}_{4}\). The permutation 1432 is vexillary and has Rothe diagram D_{1432} = {(2,2),(2,3),(3,2)} and essential set Ess(1432) = {(2,3),(3,2)}. The Rothe diagram can be visualized as follows: We draw a 4 × 4 grid and place a bullet in position (ω_{i},i) for each i. For each bullet in the grid, we draw a segment starting from it and ending on the right side of the grid and one starting from the bullet and ending on the bottom of the grid. Then D_{1432} is the set of boxes in the grid without a bullet in them or a segment through them. The elements of D_{1432} appear in gray in the figure and the elements of Ess(1432) are the lower outside corners of the Rothe diagram, that is, the boxes in the Rothe diagram so that neither the box on their right nor the box below them belongs to the Rothe diagram.
Finally, the rank function can be easily read off the above figure as follows: r_{1432}(i, j) is the number of bullets which are contained in the topleft justified subgrid of size i × j. For example, from the figure above one sees that r_{1432}(i, j) = 1 if i + j ≤ 5, r_{1432}(i, j) = 2 if i + j = 6, r_{1432}(i, j) = 3 if i + j = 7, and r_{1432}(4,4) = 4.
Definition 5.4
Let \(\omega \in \mathcal {S}_{n}\). The Schubert determinantal ideal associated to ω is
The matrix Schubert variety associated with ω is the corresponding affine variety, i.e.,
Notice that Schubert determinantal ideals are \(\mathbb {Z}^{n}\)graded, since the minors that generate them are. Moreover, by [16, Lemma 3.10] we have that
Let
be the onesided subladder of X whose lower outside corners are the elements of the essential set of ω. Y_{ω} is the set of variables of X that appear in at least one of the generators of I_{ω}. Consider the ideal generated in K[Y_{ω}] by the minors that generate I_{ω}, that is, consider \(I_{\omega }\cap K[Y_{\omega }]\subseteq K[Y_{\omega }]\). Then I_{ω} ∩ K[Y_{ω}] is \(\mathbb {Z}^{\nu }\)graded, where \(\nu =\max \limits \{j ~~ (i,j)\in \text {Ess}(\omega )\text { for some } i\}\) is the number of columns of Y_{ω}.
The family of Schubert determinantal ideals contains that of onesided ladder determinantal ideals. More precisely, consider mixed onesided ladder determinantal ideals. These are a generalization of the classical onesided ladder determinantal ideals, where the ladder can have corners in the same row or column and we take minors of different sizes in different regions of the ladder, see e.g. [19, Definition 1.4]. In [16, Proposition 9.6] it is shown that the family of mixed onesided ladder determinantal ideals coincides with that of Schubert determinantal ideals associated to vexillary permutations. Every permutation is vexillary for n ≤ 3 and the only nonvexillary permutation in \(\mathcal {S}_{4}\) is 2143. However, for large n, the proportion of vexillary permutations tends to zero as n tends to infinity [25]. Therefore, for large enough n, (mixed) onesided ladder determinantal ideals are a small subset of Schubert determinantal ideals.
Example 5.5
Consider the permutation \(\omega =1432\in \mathcal {S}_{4}\) from Example 5.3. Its Schubert determinantal ideal is I_{1432} = I_{2}(X_{2×3}) + I_{2}(X_{3×2}) = I_{2}(Y_{1432}) where Y_{1432} = X_{2×3} ∪ X_{3×2} is the subladder of X_{3×3} consisting of its first two rows and columns.
The ideal \(I_{1432}\subseteq K[x_{ij} ~~ 1\leq i,j\leq 4]\) is \(\mathbb {Z}^{4}\)graded with respect to the grading induced by letting \(\deg (x_{ij})=e_{j}\in \mathbb {Z}^{4}\). One can also regard I_{1432} as an ideal in K[Y_{1432}] = K[x_{ij}  1 ≤ i, j ≤ 3,(i, j)≠(3,3)], which is \(\mathbb {Z}^{3}\)graded graded with respect to the grading induced by letting \(\deg (x_{ij})=e_{j}\in \mathbb {Z}^{3}\).
The next result follows by combining a recent result by Fink, Mészáros, and St. Dizier [15] with results by Knutson and Miller [24] and by Brion [4]. It characterizes the Schubert determinantal ideals which are Cartwright–Sturmfels.
Theorem 5.6
Assume that K is algebraically closed. Let \(\omega \in \mathcal {S}_{n}\) and let \(I_{\omega }\subseteq S\) be the associated Schubert determinantal ideal. The following are equivalent:

(1)
I_{ω} ∈ CS(S),

(2)
I_{ω} ∩ K[Y_{ω}] ∈ CS(K[Y_{ω}]),

(3)
ω avoids the permutations 12543, 13254, 13524, 13542, 21543, 125364, 125634, 215364, 215634, 315264, 315624, and 315642.
If this is the case, then the multigraded generic initial ideal and all the initial ideals of I_{ω} are Cohen–Macaulay. Moreover, I_{ω} has a universal Gröbner basis consisting of elements of multidegree \(\leq (1,\ldots ,1,0,\ldots ,0)\in \mathbb {Z}^{n}\), where the number of ones appearing in the vector is equal to the number of columns of Y_{ω}.
Proof
By [15, Theorem 4.8], the Schubert polynomial of ω is multiplicityfree if and only if ω avoids the permutations 12543, 13254, 13524, 13542, 21543, 125364, 125634, 215364, 215634, 315264, 315624, and 315642. Moreover, the Schubert polynomial of ω coincides with the dual multidegree of S/I_{ω} by [24, Theorem A]. Therefore, ω avoids the 12 permutations listed above if and only if the only coefficients appearing in the dual multidegree of S/I_{ω} are zero and one. Notice moreover that S/I_{ω} and K[Y_{ω}]/I_{ω} ∩ K[Y_{ω}] have the same dual multidegree.
If I_{ω} is Cartwright–Sturmfels, then S/I_{ω} and K[Y_{ω}]/I_{ω} ∩ K[Y_{ω}] have multiplicityfree multidegree by Proposition 3.6. Moreover, the ideal \(I_{\omega }\subseteq S\) is prime by [16, Proposition 3.3]. In [10, Lemma 2.3], we discussed the relation between the multidegree Deg_{M}(Z) and the dual multidegree \(\text {Deg}^{\ast }_{M}(Z)\). In particular we showed that, under our assumptions, they are two different encodings of the same numerical data. In particular, Deg_{M}(Z) is multiplicityfree if and only if the only coefficients in \(\text {Deg}^{\ast }_{M}(Z)\) are zero and one. This proves that (1) implies (3) and (2) implies (3).
Conversely, suppose that (3) holds. Then S/I_{ω} and K[Y_{ω}]/I_{ω} ∩ K[Y_{ω}] have multiplicityfree multidegrees by [10, Lemma 2.3]. Since \(I_{\omega }\subseteq S\) is prime, the multigraded generic initial ideal of I_{ω} is radical and Cohen–Macaulay by [4, Theorem 1] (see also [10, Theorem 1.11] for a formulation in our terminology). This proves (1). The same argument proves (2). The rest of the statement follows from Proposition 3.4. □
The next result follows by combining Theorem 5.6 and Proposition 3.5.
Corollary 5.7
Let S = K[x_{ij}  1 ≤ j ≤ n, 0 ≤ i ≤ m_{j}] be endowed with the standard \(\mathbb {Z}^{n}\)grading induced by \(\deg (x_{ij})=e_{j}\in \mathbb {Z}^{n}\) and assume that K is algebraically closed. Let A = (a_{ij}) be an n × n matrix whose entries are \(\mathbb {Z}^{n}\)multigraded with \(\deg (a_{ij})=e_{j}\in \mathbb {Z}^{n}\) for all i, j. Let \(\omega \in \mathcal {S}_{n}\) and assume that ω avoids the permutations 12543, 13254, 13524, 13542, 21543, 125364, 125634, 215364, 215634, 315264, 315624, and 315642. Let
Then I_{ω}(A) is Cartwright–Sturmfels.
In [20] Hamaker, Pechenik, and Weigandt study the following system of generators for Schubert determinantal ideals.
Definition 5.8
Let \(X^{\prime }\) be the matrix obtained from X = (x_{ij}) by specializing x_{ij} to 0 whenever r_{ω}(i, j) = 0. The CDG generators of I_{ω} are the elements of the set
In their paper, Hamaker, Pechenik, and Weigandt formulate the following conjecture, which was later proved by Klein in [23]. We recall that a diagonal Gröbner basis is a Gröbner basis with respect to a diagonal term order, that is, a term order that selects the product of the elements on the main diagonals of the minors as initial monomials.
Theorem 5.9 (Conjecture 7.1 in 20, Corollaries 3.17 and 4.2 in [23]))
Let \(\omega \in \mathcal {S}_{n}\). The CDG generators are a diagonal Gröbner basis for I_{ω} if and only if ω avoids the permutations 13254, 21543, 214635, 215364, 215634, 241635, 315264, and 4261735.
Combining Theorems 5.6 and 5.9, one obtains the following immediate corollary.
Corollary 5.10
Let \(\omega \in \mathcal {S}_{n}\). If I_{ω} is Cartwright–Sturmfels, then the CDG generators are a diagonal Gröbner basis.
Proof
Notice that the permutations 214635 and 241635 contain 13524 and the permutation 4261735 contains 315624. Therefore, if ω avoids the list of permutations in the statement of Theorem 5.6, then it also avoids the permutations listed in Theorem 5.9. □
By comparing the lists of permutations in the statements of Theorems 5.6 and 5.9, one sees immediately that there are Schubert determinantal ideals which are not Cartwright–Sturmfels, but whose CDG generators are a diagonal Gröbner basis.
Example 5.11
Let \(\omega =13524\in \mathcal {S}_{5}\) and let I_{ω} = I_{2}(X_{2×3}) + I_{3}(X_{4×3}) be the associated Schubert determinantal ideal. Since the generators of I_{ω} are minors of Y_{ω} = X_{4×3}, we may replace X by X_{4×3} and let S = K[X_{4×3}] = K[x_{ij}∣1 ≤ i ≤ 4,1 ≤ j ≤ 3]. In particular, S and I_{ω} are \(\mathbb {Z}^{3}\)graded by letting \(\deg (x_{ij})=e_{j}\in \mathbb {Z}^{3}\).
The dual multidegree of S/I_{ω} is \({Z_{1}^{2}} Z_{2} + {Z_{1}^{2}} Z_{3} + Z_{1} {Z_{2}^{2}} + 2 Z_{1} Z_{2} Z_{3} + Z_{1} {Z_{3}^{2}} + {Z_{2}^{2}} Z_{3} + Z_{2} {Z_{3}^{2}}\), in particular it is not multiplicityfree, so I_{ω} is not Cartwright–Sturmfels.
The CDG generators of I_{ω} are the 2minors of X_{2×3} and the 3minors of X_{4×3}. Fix the lexicographic order with x_{ij} > x_{kℓ} if either i < k or i = k and j < ℓ. This is a diagonal term order. One can check by direct computation that the CDG generators are a Gröbner basis of I_{ω}.
Binomial Edge Ideals
The next theorem appeared first as [9, Theorem 2.1]. Giulia Gaggero [18] pointed out to us that the proof given in [9] contains a mistake. Indeed the equation
that is used in [9, p. 242] is not correct. The problem comes from the fact that in the proof we treated the F_{ij} as if they were the 2minors of the matrix ϕ(X) (notations as in [9]) but that it is true only up to a scalar that has been used to make them monic, hence the mistake. Here we present a correct and somehow simpler proof of [9, Theorem 2.1].
Let us set up the notation. Let G be a graph on the vertex set \(\{1,\dots , n\}\) and let X be the 2 × n matrix of variables
Denote by Δ_{ij} the 2minor of X corresponding to columns i, j, i.e., Δ_{ij} = x_{i}y_{j} − x_{j}y_{i}. We consider the binomial edge ideal of G
of \(S=K[x_{1},\dots , x_{n}, y_{1},\dots , y_{n}]\). Binomial edge ideals were introduced in [22] and [28]. We consider the \(\mathbb {Z}^{n}\)graded structure on S induced by letting \(\deg (x_{i})=\deg (y_{i})=e_{i}\in \mathbb {Z}^{n}\).
Theorem 6.1
([9, Theorem 2.1]) The multigraded generic initial ideal of J_{G} is generated by the monomials \(y_{a_{1}}{\cdots } y_{a_{v}}x_{i} x_{j}\), where i, a_{1},⋯ ,a_{v},j is a path in G. In particular J_{G} is a Cartwright–Sturmfels ideal, therefore all the initial ideals of J_{G} are radical and reg(J_{G}) ≤ n.
Proof
Consider any term order such that x_{i} > y_{i} for all i. To compute the generic initial ideal, we first apply a multigraded upper triangular transformation ϕ to J_{G}, i.e., for every i we have ϕ(x_{i}) = x_{i} and ϕ(y_{i}) = α_{i}x_{i} + y_{i} with α_{i} ∈ K. We obtain the matrix
whose 2minors are
Assume that α_{j}≠α_{i} for i≠j. We multiply ϕ(Δ_{ij}) by the inverse of α_{j} − α_{i} and obtain
with
so that F_{ij} is monic. For later reference, we observe the following: for indices 1 ≤ i < j < k ≤ n, consider the Spolynomial S(F_{ik},F_{jk}). Expanding S(F_{ik},F_{jk}) we have
Performing division with reminder by F_{ik}, F_{jk}, F_{ij} we obtain
The remainder r is
that is,
Using the syzygy among minors
we have
and
which can be checked by direct computation. Hence r = 0 and
Now we return to the ideal J_{G} and its image under ϕ:
Set
where
It suffices to prove that F is a Gröbner basis for ϕ(J_{G}), for every ϕ such that α_{j}≠α_{i} for i≠j. We first observe that \(F\subseteq \phi (J_{G})\), i.e., y_{a}F_{ij} ∈ ϕ(J_{G}) for every path i, a_{1},…,a_{v},j in G. Since F_{ij} and ϕ(Δ_{ij}) differ only by a nonzero scalar, we may as well prove that y_{a}ϕ(Δ_{ij}) ∈ ϕ(J_{G}) for every path i, a_{1},…,a_{v},j in G. This is proved easily by induction on v, the case v = 0 being trivial, applying to the matrix ϕ(X) the relation
that holds for every 2 × n matrix Z = (z_{ij}) and every triplet of column indices i, j, k. In order to prove that F is a Gröbner basis, we take two elements y_{a}F_{ij} and y_{b}F_{hk} in F and prove that the corresponding Spolynomial reduces to 0 via F. Here a = a_{1},…,a_{v} and b = b_{1}…,b_{r} and i, a, j and h, b, k are paths in G. We distinguish three cases:
Case 1. If {i, j} = {h, k}, we may assume i = h and j = k. The corresponding Spolynomial is 0.
Case 2. If {i, j}∩{h, k} = ∅. Let u = GCD(y_{a},y_{b}). Then y_{a}F_{ij} = u(y_{a}/u)F_{ij} and y_{b}F_{hk} = u(y_{b}/u)F_{hk}. Notice that (y_{a}/u)F_{ij} and (y_{b}/u)F_{hk} have coprime leading terms, hence they form a Gröbner basis. If a Gröbner basis is multiplied with a single polynomial, the resulting set of polynomials is still a Gröbner basis. Hence {y_{a}F_{ij},y_{b}F_{hk}} is a Gröbner basis and the Spolynomial of y_{a}F_{ij}, y_{b}F_{hk} reduces to 0 using only y_{a}F_{ij}, y_{b}F_{hk}.
Case 3. If {i, j}∩{h, k} = 1. Up to permuting the columns of X, we may assume that i = 1, h = 2 and j = k = n. Let u = LCM(y_{a},y_{b}). We have
Using (6.1) with i = 1, j = 2 and k = n, and multiplying both sides by u, we obtain
Since (6.1) is a division with reminder 0 of S(F_{1n},F_{2n}) with respect to F_{1n}, F_{2n}, F_{12}, we may conclude that (6.2) is a division with reminder 0 of S(y_{a}F_{1n},y_{b}F_{2n}) with respect to the set F, provided that y_{2}uF_{1n}, y_{1}uF_{2n} and y_{n}uF_{12} are multiples of elements of F. Clearly y_{2}uF_{1n} is a monomial multiple of y_{a}F_{1n} and y_{1}uF_{2n} is a monomial multiple of y_{b}F_{1n}. So we are left with y_{n}uF_{12}. If u is divisible by a monomial \(y_{d}=y_{d_{1}}{\cdots } y_{d_{t}}\) such that \(1,d_{1},\dots ,d_{t},2\) is a path in G, then y_{n}uF_{12} is a multiple of y_{d}F_{12} ∈ F. On the other hand, if u is not divisible by a monomial \(y_{d}=y_{d_{1}}{\cdots } y_{d_{t}}\) such that \(1,d_{1},\dots ,d_{t},2\) is a path in G, then
In this case, 1,a, n, b,2 is a path from 1 to 2 in G, hence y_{n}uF_{12} = y_{n}y_{a}y_{b}F_{12} ∈ F.
This concludes the proof that the set F is a Gröbner basis. The rest of the statement now follows from Proposition 3.4. □
Multigraded Closures of Linear Spaces
We now return to the notation of Section 2, in particular we let S = K[x_{ij}  1 ≤ j ≤ n, 0 ≤ i ≤ m_{j}] with the standard \(\mathbb {Z}^{n}\)grading induced by \(\deg (x_{ij})=e_{j}\).
Let \(T=K[x_{ij} ~~ 1\leq j\leq n,\ 1\leq i \leq m_{j}]\subseteq S\). Given a nonzero polynomial f ∈ T we use the variables \(x_{01},x_{02},\dots , x_{0n}\) to transform f into a polynomial of S which is \(\mathbb {Z}^{n}\)graded in a “minimal” way. Explicitly, let \(f={\sum }_{i=1}^{r} \lambda _{i} w_{i} \in T\setminus 0\) where λ_{i} ∈ K ∖{0} and w_{i} is a monomial of degree \(b_{i}=(b_{i1}, \dots , b_{in}) \in \mathbb {Z}^{n}\). Let \(d=(d_{1},\dotsm d_{n})\) with \(d_{j}=\max \limits \{ b_{1j}, \dots , b_{rj} \}\). Then the \(\mathbb {Z}^{n}\)homogenization \(f^{\hom }\in S\) of f is defined as
Notice that \(f^{\hom }\) is \(\mathbb {Z}^{n}\)homogeneous of degree \(d\in \mathbb {Z}^{n}\).
Given an ideal \(I\subseteq T\), its multigraded homogenization is the \(\mathbb {Z}^{n}\)graded ideal of S
Geometrically \(I^{\hom }\) corresponds to the closure in \(\mathbb {P}^{(m_{1},\dots , m_{n})}\) of the affine variety defined by I.
We denote by I^{⋆} the largest \(\mathbb {Z}^{n}\)graded ideal of T contained in I, i.e., the ideal generated by the \(\mathbb {Z}^{n}\)graded elements of I.
Theorem 7.1
([9, Theorem 3.1]) Let J be an ideal of T generated by homogeneous polynomials of degree 1 with respect to the \(\mathbb {Z}\)graded structure. Then \(J^{\hom }\) and J^{⋆} are Cartwright–Sturmfels ideals.
Remark 7.2
Theorem 7.1 was inspired by work of Ardila and Boocher. In their paper [2], they consider the situation m_{1} = ⋯ = m_{n} = 1. Our result recovers and generalises some of their results. Indeed the case treated by Ardila and Boocher is special, in the sense that the ideal \(J^{\hom }\) is not only a Cartwright–Sturmfels ideal but also Cartwright–Sturmfels^{∗}, a dual notion that is discussed in [10]. One important consequence of this fact is that the multigraded Betti numbers of J equal the multigraded Betti numbers of any \(\mathbb {Z}^{n}\)graded ideal with the same multigraded Hilbert function as J. In addition, any minimal multigraded system of generators is a universal Gröbner basis of J.
Example 7.3
Let n = 3 and m_{1} = m_{2} = m_{3} = 4. We consider J = (x_{i1} + x_{i2} + x_{i3} : i ∈ [4]). With
we observe that
hence \(I_{3}(X)\subseteq J\). Since I_{3}(X) is \(\mathbb {Z}^{3}\)graded we have also \(I_{3}(X) \subseteq J^{\star }\). It turns out that actually one has I_{3}(X) = J^{⋆}. This example can be generalised, see [9, Example 5.2.] where the result is presented with the transposed graded convention, i.e., with respect to the graded structure induced by \(\deg (x_{ij})=e_{i}\). Summing up, one has that for every m ≥ n and X = (x_{ij}) matrix of variables with \(\deg (x_{ij})=e_{j}\), the ideal I_{n}(X) of maximal minors of X is equal to J^{⋆} where \(J=({\sum }_{j=1}^{n} x_{ij} : i=1,\dots , m)\).
The ideals generated by linear forms are the only \(\mathbb {Z}\)graded Cartwright–Sturmfels ideals. Hence, Theorem 7.1 could be a special instance of a more general fact, that we formulate as a question.
Question 7.4
Let I be a Cartwright–Sturmfels \(\mathbb {Z}^{n}\)graded ideal of S. Suppose that we introduce a finer graded structure on S, say a \(\mathbb {Z}^{r}\)graded structure with r > n such that if two variables have the same \(\mathbb {Z}^{r}\)degree then they have the same \(\mathbb {Z}^{n}\)degree. Then I is not necessarily \(\mathbb {Z}^{r}\)graded and we may consider its \(\mathbb {Z}^{r}\)homogenization \(I^{\hom }\subseteq S[y_{1},\dots , y_{r}]\) and homogeneous \(\mathbb {Z}^{r}\)part I^{∗}. Are \(I^{\hom }\) and I^{∗} Cartwright–Sturmfels ideals?
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The first and the second authors were partially supported by GNSAGAINdAM.
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Dedicated to our friend and colleague Bernd Sturmfels on the occasion of his sixtieth birthday.
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Conca, A., De Negri, E. & Gorla, E. Radical Generic Initial Ideals. Vietnam J. Math. 50, 807–827 (2022). https://doi.org/10.1007/s1001302200551w
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DOI: https://doi.org/10.1007/s1001302200551w
Keywords
 Cartwright–Sturmfels ideals
 Determinantal ideals
 Radical ideals
 Multidegrees
Mathematics Subject Classification (2010)
 Primary 13C40
 13P10
 05E40
 Secondary 14M99