Let $X$ be an $n\times m$ matrix of indeterminates over a field $K$ (of sufficiently large characteristic) and $M_t$ the set of $m$-minors of $X$. We consider two objects: (1) the Ress algebra of the polynomial ring $K[X]$ with respect to the ideal $I_t$ generated by $M_t$, and (2) the $A_t$ subalgebra of $K[X]$ generated by $M_t$. Note that $A_t$ is tHE coordinate ring of a Grassmannian if $t=\min(m,n)$; also the cases $t=1$ and $t=m-1=n-1$ are easily understood, since $A_t$ is a polynomial ring over $K$ in these cases.
For both objects we compute the divisor class group and the canonical class. In particular we determine the Gorenstein rings among the $A_t$. It turns out that $A_t$ is Gorenstein exactly in the cases listed above and when $t(m+n)=mn$.
We use initial methods, based on the straightening law and KRS. They can be applied to other types of determinantal ideals, too. We do this explicitly for generic Hankel matrices.
We give a description of the minimal primes of the ideal generated by the 2 x 2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m x m minors of an m x n generic matrix when the characteristic of the ground field is zero. A key intermediate result is the proof that the ideals which appear as minimal primes are, in fact, prime ideals. This introduces a large new class of mixed determinantal ideals that are prime.
This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The main result in the first part of this paper is a theorem which gives sufficient conditions for the ideal of minors of a matrix to be prime. This theorem is general enough to include interesting examples, such as the ideal of maximal minors of catalecticant matrices and their generalisations discussed in the second part of the paper. The second aim of this paper is to settl…
We study higher jumping numbers and generalized test ideals associated to determinantal ideals over a field of positive characteristic. We work in positive characteristic and give a complete characterization of both families for ideals generated by maximal minors; and make a conjecture for the general case. Although these invariants are understood asymptotically - as their positive characteristic grows to infinity they essentially coincide with their characteristic zero analogues - there are known examples where the behavior in positive characteristic differs significantly from that of charact…
The classical "generalized principal ideal theorems" of Macaulay, Eagon-Northcott, and others give sharp bounds on the heights of determinantal ideals in arbitrary rings. But in regular local rings (or graded polynomial rings) these are far from sharp, and various questions about vector bundles, as well as other questions in commutative algebra, amount to asking what the real bounds are. We give partial answers, and more generally we prove new height bounds in local rings of given embedding codimension. Our theorems extend and sharpen results of Faltings and Bruns.
Ideals generated by adjacent 2-minors are studied. First, the problem when such an ideal is a prime ideal as well as the problem when such an ideal possesses a quadratic Gröbner basis is solved. Second, we describe explicitly a primary decomposition of the radical ideal of an ideal generated by adjacent 2-minors, and challenge the question of classifying all ideals generated by adjacent 2-minors which are radical ideals. Finally, we discuss connectedness of contingency tables in algebraic statistics.
We compute the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix, as well as of those with support in the ideal of 2n x 2n Pfaffians of a (2n+1)x(2n+1) generic skew-symmetric matrix. As an application, we characterize the Cohen-Macaulay modules of covariants for the action of the special linear group SL(G) on G^m. The main tool we develop is a method for computing certain Ext modules based on the geometric technique for computing syzygies and on Matlis duality.
An ideal I in a polynomial ring S has linear powers if all the powers I^k of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that every rational normal scroll has linear powers.
We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory.
Suppose that k is a field of characteristic zero, X is an r by s matrix of indeterminates, where r \leq s, and R = k[X] is the polynomial ring over k in the entries of X. We study the local cohomology modules H^i_I(R), where I is the ideal of R generated by the maximal minors of X. We identify the indices i for which these modules vanish, compute H^i_I(R) at the highest nonvanishing index, i = r(s-r)+1, and characterize all nonzero ones as submodules of certain indecomposable injective modules. These results are consequences of more general theorems regarding linearly reductive groups acting o…
Secant varieties to Veronese embeddings of projective space are classical varieties whose equations are not completely understood. Minors of catalecticant matrices furnish some of their equations, and in some situations even generate their ideals. Geramita conjectured that this is the case for the secant line variety of the Veronese variety, namely that its ideal is generated by the 3x3 minors of any of the "middle" catalecticants. Part of this conjecture is the statement that the ideals of 3x3 minors are equal for most catalecticants, and this was known to hold set-theoretically. We prove the…
We compute the primary decomposition of certain ideals generated by subsets of minors in a generic matrix or in a generic symmetric matrix, or subsets of Pfaffians in a generic skew-symmetric matrix. Specifically, the ideals we consider are generated by minors that have at least some given number of rows and columns in certain submatrices.
We compute the local cohomology modules H_Y^(X,O_X) in the case when X is the complex vector space of n x n symmetric, respectively skew-symmetric matrices, and Y is the closure of the GL-orbit consisting of matrices of any fixed rank, for the natural action of the general linear group GL on X. We describe the D-module composition factors of the local cohomology modules, and compute their multiplicities explicitly in terms of generalized binomial coefficients. One consequence of our work is a formula for the cohomological dimension of ideals of even minors of a generic symmetric matrix: in the…
In this paper we study ideals generated by quite general sets of 2-minors of an $m \times n$-matrix of indeterminates. The sets of 2-minors are defined by collections of cells and include 2-sided ladders. For convex collections of cells it is shown that the attached ideal of 2-minors is a Cohen--Macaulay prime ideal. Primality is also shown for collections of cells whose connected components are row or column convex. Finally the class group of the ring attached to a stack polyomino and its canonical class is computed, and a classification of the Gorenstein stack polyominoes is given.
We study a large family of products of Borel fixed ideals of maximal minors. We compute their initial ideals and primary decompositions, and show that they have linear free resolutions. The main tools are an extension of straightening law and a very surprising primary decomposition formula. We study also the homological properties of associated multi-Rees algebra which are shown to be Cohen-Macaulay, Koszul and defined by a Gröbner basis of quadrics.
We study algebraic and homological properties of the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all $(n-1)$-minors of a symmetric $n \times n$ matrix whose entries in the upper triangle are distinct variables or zeros and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded as a simple graph $G$ with $n$ vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute al…
All matrices we consider have entries in a fixed algebraically closed field $K$. A minor of a square matrix is principal means it is defined by the same row and column indices. We study the ideal generated by size $t$ principal minors of a generic matrix, and restrict our attention to locally closed subsets of its vanishing set, given by matrices of a fixed rank. The main result is a computation of the dimension of the locally closed set of $n\times n$ rank $n-2$ matrices whose size $n-2$ principal minors vanish; this set has dimension $n^2-n-4$.
We show that the ideal generated by the $(n-2)$ minors of a general symmetric $n$ by $n$ matrix has an initial ideal that is the Stanley-Reisner ideal of the boundary complex of a simplicial polytope and has the same Betti numbers.
With a simple graph $G$ on $[n]$, we associate a binomial ideal $P_G$ generated by diagonal minors of an $n \times n$ matrix $X=(x_{ij})$ of variables. We show that for any graph $G$, $P_G$ is a prime complete intersection ideal and determine the divisor class group of $K[X]/ P_G$. By using these ideals, one may find a normal domain with free divisor class group of any given rank.
Let G be a finite graph on [n] = {1,2,3,...,n}, X a 2 times n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this paper, we study about ideals I_G of S generated by 2-minors [i,j] of X which correspond to edges {i,j} of G. In particular, we construct a Groebner basis of I_G as a set of paths of G and compute a primary decomposition.
We describe the universal Groebner basis of the ideal of maximal minors and the ideal of $2$-minors of a multigraded matrix of linear forms. Our results imply that the ideals are radical and provide bounds on the regularity. In particular, the ideals of maximal minors have linear resolutions. Our main theoretical contribution consists of introducing two new classes of ideals named after Cartwright and Sturmfels, and proving that they are closed under multigraded hyperplane sections. The gins of the ideals that we study enjoy special properties.
In arXiv:math/0405373 , Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of the embedding of a projective space $\mathbb{P}^{n-1}\hookrightarrow \mathbb{P}^{r-1}$ determined by generators of a linearly presented $\mathfrak{m}$-primary ideal. This result implies in particular that the image is scheme defined by equations of degree at most $n$. In this text we prove that the ideal of maximal minors of the Jacobian dual matrix associated to the input ideal defines the image as a scheme; it is generated in degree $n$. Showing that this ideal has a linear resol…
Cut ideals are used in algebraic statistics to study statistical models defined by graphs. Intuitively, topological restrictions on the graphs should imply structural statements about the corresponding cut ideals. Several theorems and many computer calculations support that. Sturmfels and Sullivant conjectured that the cut ideal is generated by quadrics if and only if the graph is free of K4-minors. Parts of the conjecture has been resolved by Brennan and Chen, and later by Nagel and Petrovic. We prove the full conjecture by introducing a new type of toric fiber product theorem.
A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$ denote the ideal generated by the size $t$ principal minors of $X$. When $t=2$ the resulting quotient ring $K[X]/\mathfrak P_2$ is a normal complete intersection domain. When $t>2$ we break the problem into cases depending on a fixed rank, $r$, of $X$. We show when $r=n$ for any $t$, the respective images of $\mathfrak P_t$ and $\mathfrak P_{n-t}$ in the lo…
We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of other ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the Rees rings of a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, …
Let $m$, $n$, $a_1$, ..., $a_r$, $b_1$, ..., $b_r$ be integers with $1\leq a_1<...<a_r\leq m$ and $1\leq b_1<...<b_r\leq n$. And let $x$ be the universal $m\times n$ matrix with the property that $i$-minors of first $a_i-1$ rows and first $b_i-1$ columns are all zero, for $i=1$, ..., $r+1$ ($a_{r+1}=m+1$ and $b_{r+1}=n+1$). For an integer $u$ with $1\leq u\leq m$, we denote by $U$ the $u\times n$ matrix consisting of the first $u$ rows of $x$. In this paper, we consider the analytic spread and the reduction number of the ideal of maximal minors of $U$
Consider the affine space consisting of pairs of matrices $(A,B)$ of fixed size, and its closed subvariety given by the rank conditions $\operatorname{rank} A \leq a$, $\operatorname{rank} B \leq b$ and $\operatorname{rank} (A\cdot B) \leq c$, for three non-negative integers $a,b,c$. These varieties are precisely the orbit closures of representations for the equioriented $A_3$ quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of…
Let $G$ be a simple graph on the vertex set $\{1,\ldots,n\}$ with $m$ edges. An algebraic object attached to $G$ is the ideal $P_{G}$ generated by diagonal 2-minors of an $n \times n$ matrix of variables. In this paper we prove that if $G$ is bipartite, then every initial ideal of $P_{G}$ is generated by squarefree monomials of degree at most $\left \lfloor{\frac{m+n+1}{2}} \right \rfloor$. Furthermore, we completely characterize all connected graphs $G$ for which $P_{G}$ is the toric ideal associated to a finite simple graph. Finally we compute in certain cases the universal Gr{ö}bner basis o…
Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In two previous papers we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series.
For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant description of the local cohomology modules of the polynomial ring S of functions on the space of m x n matrices, with support in the ideal of p x p minors. Our techniques allow us to explicitly compute all the modules Ext_S(S/I_x,S), for x a partition and I_x the ideal generated by the irreducible sub-representation of S indexed by x. In particular we determine the regularity of the ideals I_x, and we deduce that the only ones admitting a linear free resolution are the powers of the ideal of maximal minors of the generi…
We show that the ideal generated by maximal minors (i.e., $(k+1)$-minors) of a $(k+1) \times n$ Vandermonde matrix is radical and Cohen-Macaulay. Note that this ideal is generated by all Specht polynomials with shape $(n-k,1,...,1)$.
We determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m x n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.
We consider ideals generated by general sets of $m$-minors of an $m\times n$-matrix of indeterminates. The generators are identified with the facets of an $(m-1)$-dimensional pure simplicial complex. The ideal generated by the minors corresponding to the facets of such a complex is called a determinantal facet ideal. Given a pure simplicial complex $Δ$, we discuss the question when the generating minors of its determinantal facet ideal $J_Δ$ form a Gröbner basis and when $J_Δ$ is a prime ideal.
In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators of $I_{\mathcal M}$. In particular, given a minimal set of binomial generators of $I_{\mathcal M}$ we provide a necessary condition for $\mathcal M$ to have a minor isomorphic to $\mathcal U_{d,2d}$ for $d \geq 2$. This condition is proved to be suff…
We introduce the concept of s-Hankel hypermatrix, which generalizes both Hankel matrices and generic hypermatrices. We study two determinantal ideals associated to an s-Hankel hypermatrix: the ideal I<s,t> generated by certain 2 x 2 slice minors, and the ideal \tilde{I}<s,t> generated by certain 2 x 2 generalized minors. We describe the structure of these two ideals, with particular attention to the primary decomposition of I<s,t>, and provide the explicit list of minimal primes for large values of s. Finally we give some geometrical interpretations and generalise a theorem of Watanabe.
It is well-known that the Plücker relations generate the ideal of relations of the maximal minors of a generic matrix. In this paper we discuss the relations between minors of a (non-maximal) fixed size. We will exhibit minimal relations in degrees 2 (non-Plücker in general) and 3, and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is through the representation theory of the general linear group.
The vector space of m x n complex matrices (m >= n) admits a natural action of the group GL = GL_m x GL_n via row and column operations. For positive integers a,b, we consider the ideal I_{a x b} defined as the smallest GL-equivariant ideal containing the b-th powers of the a x a minors of the generic m x n matrix. We compute the syzygies of the ideals I_{a x b} for all a,b, together with their GL-equivariant structure, generalizing earlier results of Lascoux for the ideals of minors (b=1), and of Akin-Buchsbaum-Weyman for the powers of the ideals of maximal minors (a=n). Our methods rely on a…
A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix where $n\leq m$ indexed by the facets of a simplicial complex $Δ$. We consider the more general notion of an $r$-DFI, which is generated by a subset of $r$-minors of a generic matrix indexed by the facets of $Δ$ for some $1\leq r\leq n$. We define and study so-called lcm-closed and unit interval $r$-DFIs, and show that the minors parametrized by the facets of $Δ$ form a reduced Gröbner basis with respect to \emph{any} term order for an lcm-closed $r$-DFI. We also see that being lcm-…
We compute the deformation space of quadratic letterplace ideals $L(2,P)$ of finite posets $P$ when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal $J(2,P)$ defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases $J(2,P)$ is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.
A set of polynomials G in a polynomial ring S over a field is said to be a universal Groebner basis, if G is a Groebner basis with respect to every term order on S. Twenty years ago Bernstein, Sturmfels, and Zelevinsky proved that the set of the maximal minors of a matrix X of variables is a universal Groebner basis. Boocher recently proved that any initial ideal of the ideal of maximal minors of X has a linear resolution. In this paper we give a quick proof of the results mentioned above. Our proof is based on a specialization argument. Then we show that similar statements hold in a more g…
We introduce a class of ideals generated by a set of 2-minors of $m\times n$-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gröbner basis and characterize unmixedness and Cohen--Macaulayness.
In this paper, we give a sufficient condition for a set $\mathal G$ of polynomials to be a Gröbner basis with respect to a given term-order for the ideal $I$ that it generates. Our criterion depends on the linkage pattern of the ideal $I$ and of the ideal generated by the initial terms of the elements of $\mathcal G$. We then apply this criterion to ideals generated by minors and pfaffians. More precisely, we consider large families of ideals generated by minors or pfaffians in a matrix or a ladder, where the size of the minors or pfaffians is allowed to vary in different regions of the matrix…
The ideal of a Segre variety is generated by the 2-minors of a generic hypermatrix of indeterminates. We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of weak generic hypermatrix which allows us to treat also the case of projection of Veronese surfaces from a set of generic points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.
We consider determinantal ideals, where the generating minors are encoded in a hypergraph. We study when the generating minors form a Gröbner basis. In this case, the ideal is radical, and we can describe algebraic and numerical invariants of these ideals in terms of combinatorial data of their hypergraphs, such as the clique decomposition. In particular, we can construct a minimal free resolution as a tensor product of the minimal free resolution of their cliques. For several classes of hypergraphs we find a combinatorial description of the minimal primes in terms of a prime splitting. That i…
