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 $U$
Let $S=\mathbb{C}[x_{ij}]$ be a polynomial ring of $m\times n$ variables over $\mathbb{C}$ and let $I$ be the determinantal ideal of maximal minors of $S$. Using the representation theoretic techniques introduced in arXiv:1305.1719, arXiv:1309.0617 and arXiv:1611.00415, we prove the existence of the generalized $j$-multiplicities $ε^j(I)$ defined by Dao and Montaño in arXiv:1705.05033. We will also give a closed formula of $ε^j(I)$, which generalized the results in arXiv:1308.0582 and arXiv:1912.02917 in the maximal minors case.
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.
Let $S=\mathbb{C}[x_{ij}]$ be a polynomial ring of $m\times n$ generic variables (resp. a polynomial ring of $(2n+1) \times (2n+1)$ skew-symmetric variables) over $\mathbb{C}$ and let $I$ (resp. Pf) be the determinantal ideal of maximal minors (resp. sub-maximal pfaffians) of $S$. Using the representation theoretic techniques introduced in the work of Raicu et al, we study the asymptotic behavior of the length of the local cohomology module of determinantal and pfaffian thickenings for suitable choices of cohomological degrees. This asymptotic behavior is also defined as a notion of multiplict…
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.
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 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.
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…
Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay -- a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tool…
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…
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…
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.
We study the extremal Betti numbers of the class of $t$--spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for $t\in \{1,2\}$.
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…
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-…
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.
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 $(n-k,1,...,1)$.
Linear resolution of products of monomial ideals related to maximal minors
Ramakrishna Mission Vivekenanda Educational and Research Institute（拉达克里希纳使命维韦卡南达教育研究所）；Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）；Indian Institute of Technology Bhubaneswar（印度理工学院布巴内斯瓦尔分校）
In our paper "Non-commutative desingularization of determinantal varieties, I" we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper we prove the existence of non-commutative resolutions in the general case in a manner which is still characteristic free, and carry out the explicit description by generators and relations in characteristic zero. As an applica…
We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be obtained by means of a simple computer algebra procedure once the integral closure of each row ideal is known. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.
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.
Let A be a Noetherian local domain, N be a finitely generated torsion- free module, and M a proper submodule that is generically equal to N. Let A[N] be an arbitrary graded overdomain of A generated as an A-algebra by N placed in degree 1. Let A[M] be the subalgebra generated by M. Set C:=Proj(A[M]) and r:=dim C. Form the (closed) subset W of Spec(A) of primes p where A[N]_p is not a finitely generated module over A[M]_p, and denote the preimage of W in C by E. We prove this: (1) dim E=r-1 if either (a) N is free and A[N] is the symmetric algebra, or (b) W is nonempty and A is universally cate…
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…
We generalize Gaeta's Theorem to the family of determinantal schemes. In other words, we show that the schemes defined by minors of a fixed size of a matrix with polynomial entries belong to the same G-biliaison class of a complete intersection whenever they have maximal possible codimension, given the size of the matrix and of the minors that define them.
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 determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.
Institute of Mathematics, Hebrew University（希伯来大学数学研究所）；Universidad de Valladolid（巴利亚多利德大学）；Einstein Institute of Mathematics, Hebrew University（希伯来大学爱因斯坦数学研究所）
We study determinantal Cremona maps, i.e. birational maps whose base ideal is the maximal minors ideal of a given matrix $Φ$, via the resolution of the polynomials systems defined by $Φ$. Using convex geometry, this approach leads in particular to describe the projective degrees of some almost linear determinantal maps.
