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 same-size minors of the matrix of variables. Let X = (xij) be an m × n matrix of variables over a field K and S = K[xij : 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 mj = m − 1 for \(j=1,\dots , n\) and, in accordance with usual notation for matrices, the index i varies from 1 to m. Let It(X) be the ideal of S generated by the t-minors of X. Clearly It(X) is \(\mathbb {Z}^{n}\)-graded and our first goal is to compute the multidegree of S/It(X).
The multigraded Hilbert function, hence the multidegree, does not change if we replace It(X) with an initial ideal. The ideal It(X) has a well-known 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 t-minors, whose associated simplicial complex will be denoted by πt. The facets of πt can be identified with the families of non-intersecting paths in the grid [m] × [n] from the starting points p1 = (1,n),p2 = (2,n),…,pt− 1 = (t − 1,n) to the endpoints q1 = (m,1),q2 = (m,2),…,qt− 1 = (m, t − 1).
For example, for m = 4, n = 5, and t = 3, we have p1 = (1,5), p2 = (2,5), q1 = (4,1), and q2 = (4,2). The following is a facet of π3:
$$ \begin{array}{ccccccc} - & 1 & 1 & 1 &1 & \leftarrow p_{1} \\ 1 & 1 & - & 2 &2 & \leftarrow p_{2} \\ 1 & - & 2 & 2 & - \\ 1& 2 & 2 & - & - \\ \uparrow & \uparrow \\ q_{1} & q_{2} \end{array} $$
depicted using the matrix coordinates and marking with “1” the lattice points which belong to the first path (from p1 to q1) and with “2” the lattice points of the second path (from p2 to q2).
Each family of non-intersecting paths must have at least t − 1 points on each column. Therefore, each facet of πt 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
$$ \text{Deg}_{S/I_{t}(X)}(Z_{1},\dots,Z_{n})= \sum\limits_{F\in \mathbb{F}({\Pi}_{t})} Z_{1}^{c_{1}(F)-1}{\cdots} Z_{n}^{c_{n}(F)-1}, $$
(4.1)
where cj(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
$$ W(U,Z_{1},\dots, Z_{n})= \sum\limits_{F\in U} Z_{1}^{c_{1}(F)}{\cdots} Z_{n}^{c_{n}(F)}, $$
so that we may rewrite (4.1) as
$$ \text{Deg}_{S/I_{t}(X)}(Z_{1},\dots,Z_{n}) =(Z_{1}{\cdots} Z_{n})^{-1} W(\mathbb{F}({\Pi}_{t}),Z_{1},\dots, Z_{n}). $$
(4.2)
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
$$ W(\mathbb{F}({\Pi}_{t}),Z_{1},\dots, Z_{n})=\det \left( W(\text{Paths}(p_{i},q_{j}) ,Z_{1},\dots, Z_{n})\right)_{i,j=1,\dots, t-1}, $$
(4.3)
where Paths(pi,qj) is the set of the paths from pi to qj.
In the sequel, hv(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
$$ W(\text{Paths}(p,q) ,Z_{1},\dots, Z_{n})=\left( \prod\limits_{i=d}^{b} Z_{i} \right) h_{c-a}(Z_{d},Z_{d+1},\dots, Z_{b}). $$
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 cj(P) > 0 if and only if d ≤ j ≤ b and \({\sum }_{j=d}^{b} c_{j}(P)=(b-d)+(c-a)+1\). 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/It(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
$$ \text{Deg}_{S/I_{t}(X)}(Z_{1},\dots,Z_{n})=(Z_{1}{\cdots} Z_{n})^{t-2} \det\left( h_{m+1-t-i+j}(Z_{1},\dots,Z_{n})\right)_{i,j=1,2,\dots, t-1}. $$
Proof
Combining (4.2) and (4.3) with Lemma 4.1 we obtain
$$ \text{Deg}_{S/I_{t}(X)}(Z_{1},\dots,Z_{n}) = (Z_{1}{\cdots} Z_{n})^{-1} \det\left( \left( \prod\limits_{k=j}^{n} Z_{k}\right) h_{m-i}(Z_{j},\dots, Z_{n})\!\right)_{i,j=1,2,\dots, t-1}. $$
For \(j=1,\dots ,t-1\), the factor \({\prod }_{k=j}^{n} Z_{k} \) can be extracted from the determinant, hence the monomial in front of the determinant becomes
$$ Z_{2}{Z_{3}^{2}}{\cdots} Z_{t-2}^{t-3} Z_{t-1}^{t-2} Z_{t}^{t-2} {\cdots} Z_{n}^{t-2}, $$
while the determinant becomes
$$ \det\left( h_{m-i}(Z_{j},\dots, Z_{n})\right)_{i,j=1,2,\dots, t-1}. $$
It now suffices to prove that the latter equals
$$ Z_{1}^{t-2}Z_{2}^{t-3}{\cdots} Z_{t-2} \det\left( h_{m+1-t-i+j}(Z_{1},\dots, Z_{n})\right). $$
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
$$ \begin{array}{@{}rcl@{}} &&\det\left( \begin{array}{lll} h_{m-1}(Z_{1},\dots, Z_{n}) & h_{m-1}(Z_{2},\dots, Z_{n}) & h_{m-1}(Z_{3},\dots, Z_{n}) \\ h_{m-2}(Z_{1},\dots, Z_{n}) & h_{m-2}(Z_{2},\dots, Z_{n}) & h_{m-2}(Z_{3},\dots, Z_{n}) \\ h_{m-3}(Z_{1},\dots, Z_{n}) & h_{m-3}(Z_{2},\dots, Z_{n}) & h_{m-3}(Z_{3},\dots, Z_{n}) \end{array}\right)\\ &&= {Z_{1}^{2}}Z_{2}\det\left( \begin{array}{lll} h_{m-3}(Z_{1},\dots, Z_{n}) & h_{m-2}(Z_{1},\dots, Z_{n}) & h_{m-1}(Z_{1},\dots, Z_{n}) \\ h_{m-4}(Z_{1},\dots, Z_{n}) & h_{m-3}(Z_{1},\dots, Z_{n}) & h_{m-2}(Z_{1},\dots, Z_{n}) \\ h_{m-5}(Z_{1},\dots, Z_{n}) & h_{m-4}(Z_{1},\dots, Z_{n}) & h_{m-3}(Z_{1},\dots, Z_{n}) \end{array}\right). \end{array} $$
(4.4)
In the first term of (4.4), we subtract the second column from the third and the first column from the second. Since \(h_{v}(Z_{j+1},\dots , Z_{n})-h_{v}(Z_{j},\dots , Z_{n})=-Z_{j} h_{v-1}(Z_{j},\dots , Z_{n})\), we can factor out − Z2 from the third column and − Z1 from the second. The first term of (4.4) therefore becomes
$$ Z_{1}Z_{2} \det\left( \begin{array}{lll} h_{m-1}(Z_{1},\dots, Z_{n}) & h_{m-2}(Z_{1},\dots, Z_{n}) & h_{m-2}(Z_{2},\dots, Z_{n}) \\ h_{m-2}(Z_{1},\dots, Z_{n}) & h_{m-3}(Z_{1},\dots, Z_{n}) & h_{m-3}(Z_{2},\dots, Z_{n}) \\ h_{m-3}(Z_{1},\dots, Z_{n}) & h_{m-4}(Z_{2},\dots, Z_{n}) & h_{m-4}(Z_{2},\dots, Z_{n}) \end{array}\right). $$
Subtracting the second column from the third and factoring − Z1, we obtain
$$ -{Z_{1}^{2}}Z_{2}\det\left( \begin{array}{lll} h_{m-1}(Z_{1},\dots, Z_{n}) & h_{m-2}(Z_{1},\dots, Z_{n}) & h_{m-3}(Z_{1},\dots, Z_{n}) \\ h_{m-2}(Z_{1},\dots, Z_{n}) & h_{m-3}(Z_{1},\dots, Z_{n}) & h_{m-4}(Z_{1},\dots, Z_{n}) \\ h_{m-3}(Z_{1},\dots, Z_{n}) & h_{m-4}(Z_{2},\dots, Z_{n}) & h_{m-5}(Z_{1},\dots, Z_{n}) \end{array}\right). $$
Finally we exchange rows one and three and then transpose. This yields
$$ {Z_{1}^{2}}Z_{2}\det\left( \begin{array}{lll} h_{m-3}(Z_{1},\dots, Z_{n}) & h_{m-2}(Z_{1},\dots, Z_{n}) & h_{m-1}(Z_{1},\dots, Z_{n}) \\ h_{m-4}(Z_{1},\dots, Z_{n}) & h_{m-3}(Z_{1},\dots, Z_{n}) & h_{m-2}(Z_{1},\dots, Z_{n}) \\ h_{m-5}(Z_{1},\dots, Z_{n}) & h_{m-4}(Z_{1},\dots, Z_{n}) & h_{m-3}(Z_{1},\dots, Z_{n}) \end{array}\right) $$
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 non-negative integers λ1,λ2,…,λr. Given a partition λ = λ1,λ2,…,λr the Schur polynomial sλ(Z) associated λ and with respect to the variables
$$ Z=Z_{1},\dots, Z_{n} $$
is
$$ s_{\lambda}(Z)=\det \left( h_{\lambda_{i}-i+j}(Z) \right)_{i,j=1,\dots,r}. $$
By construction sλ(Z) is a symmetric homogeneous polynomial of degree \(|\lambda |={\sum }_{i=1}^{r} \lambda _{i}\) with integral coefficients. Denote by mμ(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 mμ(Z)’s form a K-basis of the space of symmetric polynomials, one can express sλ(Z) as
$$ s_{\lambda}(Z)= \sum\limits_{\mu ~:~ |\mu|=|\lambda|} K_{\lambda, \mu} m_{\mu}(Z). $$
The coefficients Kλ, 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 semi-standard (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)
Kλ, 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)
Kλ, μ is the number of semi-standard tableaux of shape λ with entries \(1,\dots ,n\) and multiplicities given by μ (i.e., μ1 entries are equal to 1, μ2 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/It(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
$$ \text{Deg}_{S/I_{t}(X)}(Z_{1},\dots,Z_{n})=(Z_{1}{\cdots} Z_{n})^{t-2} s_{\lambda}(Z) $$
where
$$ \lambda=\ell^{(t-1)}=\underbrace{\ell,\ell,\dots,\ell}_{(t-1)\text{-times}} \text{ where } \ell=m+1-t. $$
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/It(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 eb(S/It(X)) satisfy the following properties:
-
(1)
eb(S/It(X)) is a symmetric function of b.
-
(2)
eb(S/It(X)) > 0 if and only if t − 2 ≤ bi ≤ m − 1 for every \(i=1,\dots , n\).
-
(3)
Set ci = bi + 1 for \(i=1,\dots , n\). Then eb(S/It(X)) is the number of families of non-intersecting paths from \(p_{1}=(1,n), p_{2}=(2,n), \dots , p_{t-1}=(t-1,n)\) to \(q_{1}=(m,1), q_{2}=(m,2), {\dots } , q_{t-1}=(m, t-1)\) with exactly ci points on the i-th column for \(i=1,\dots , n\).
-
(4)
Set μi = bi − (t − 2) for \(i=1,\dots , n\). Then eb(S/It(X)) equals the Kostka number Kλ, μ, that is, the number of semi-standard tableaux of shape λ with entries \(1,\dots ,n\) and multiplicities given by μ.
Proof
(1) The fact that eb(S/It(X)) is a symmetric function of b follows from the fact that It(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
$$ \begin{array}{@{}rcl@{}} \text{Deg}_{R/I_{3}(X)}(Z)= z \ s_{2,2}(Z) &=& z \left( m_{(2,2,0,0)}+m_{(2, 1, 1, 0)}+2 m_{(1,1,1,1)} \right)\\ &=& m_{(3,3,1,1)}+m_{(3, 2, 2, 1)}+2 m_{(2,2,2,2)}. \end{array} $$
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 K22,1111, i.e., the number of semistandard tableaux of shape 2,2 with entries \(1,\dots ,4\) and multiplicities given by (1,1,1,1):
$$ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \quad \begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array}$$
Moreover, it is also the number of families of non-intersecting paths from p1 = (1,4), p2 = (2,4) to q1 = (4,1), q2 = (4,2) with 3 points on each column:
$$\begin{array}{cccc} - & 1 & 1 & 1 \\ 1 & 1 & - & 2 \\ 1 & - & 2 & 2 \\ 1& 2 & 2 & - \end{array} \qquad \begin{array}{cccc} - & - & 1 & 1 \\ 1 & 1 & 1 & 2 \\ 1 & 2 & 2 & 2 \\ 1& 2 & - & - \end{array}$$
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 Li 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
$$ \text{Deg}_{R/I_{n-1}(X)}(Z)= z^{n-2} \ s_{2^{(n-2)}}(Z)= z^{n-2} \left( m_{\mu_{1}}+m_{\mu_{2}}+2 m_{\mu_{3}} \right) $$
with
$$ \mu_{i}=\left\{ \begin{array}{l} (2^{(n-4)},2,2,0,0)\quad \text{ if } i=1,\\ (2^{(n-4)},2,1,1,0)\quad \text{ if } i=2,\\ (2^{(n-4)},1,1,1,1)\quad \text{ if } i=3. \end{array} \right. $$
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 It(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/It(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/It(X) has a multiplicity-free 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 eb(S/It(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 eb(S/It(X)) > 1.
Claim 1 for t = 2 follows immediately from Theorem 4.2. Claim 1 for t = m ≤ n follows from the description of eb(S/It(X)) in Theorem 4.5(4) in terms of semi-standard 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), eb(S/In(X)) > 0 if and only if n − 2 ≤ bi ≤ m − 1 and \(|b|=\dim S/I_{n}(X)-n\). Setting bi = (m − 1) − ci and rewriting the conditions with respect to \((c_{1},\dots , c_{n})\), we have that eb(S/In(X)) > 0 if and only if \((c_{1},\dots , c_{n})\in \mathbb {N}^{n}\) and \({\sum }_{i=1}^{n} c_{i}=m-n+1\). Hence, there are exactly \(\left (\begin {array}{c}{m}\\{n-1} \end {array}\right )\) elements \(b\in \mathbb {N}^{n}\) such that eb(S/In(X)) > 0. By (2.1), \({\sum }_{b} e_{b}(S/I_{n}(X))\) gives the ordinary multiplicity of S/In(X), which is \(\left (\begin {array}{c}{m}\\{n-1} \end {array}\right )\). It follows that eb(S/In(X)) = 1 whenever eb(S/In(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 semi-standard tableaux of shape ℓ(t− 1) and entries \(1,\dots , n\) with multiplicities given by μ. One can take
$$ \mu_{i}= \left\{\begin{array}{ll} \ell\quad& \text{ if } i=1,2,\dots, t-3, \\ \ell-1 \quad& \text{ if }i=t-2, t-1, \\ 1 \quad& \text{ if } i=t, t+1, \\ 0 \quad& \text{ if } i=t+2, \dots, n. \end{array} \right. $$
For \(i=1,\dots , t-3\), the i-th row of any semi-standard 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 \(j=1,\dots , m-4\) the j-th 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/It(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 It(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 It(X) is a relevant prime, the conclusion follows combining Theorem 4.9 and Proposition 3.6. □
The fact that I2(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 It(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 = (aij) be an m × n matrix whose entries are \(\mathbb {Z}^{n}\)-multigraded with \(\deg (a_{ij})=e_{j}\) for all i, j. Let It(A) be the ideal of t-minors of A. Then It(A) is Cartwright–Sturmfels for \(t=1,2,\min \limits \{m,n\}\).
In particular when \(t=1,2,\min \limits \{m,n\}\) then It(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 Im(A).
Remark 4.13
It is natural to ask whether the ideal It(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 It(A) = 0) and no for a general enough A (for example if It(A) has the expected codimension). Nevertheless, notice that the generators of It(A) have squarefree \(\mathbb {Z}^{n}\)-degrees, hence cannot have factors of multiplicity larger than one. This suggest that It(A) might always be radical. It turns out that this is not the case: In [12, Example 7.2] the authors give examples of non-radical coordinate sections of determinantal ideals.
The multidegree of S/It(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 It(A), from which the multidegree is easily derived.