We investigate the defining ideal of a set of points X in multi-projective space with a special emphasis on the case that X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, we determine the degrees of the generators of the associated ideal I_X. Letting ν(I_X) denote the minimal number of generators of I_X, we use this description of the degrees to construct a function v(s;n_1,...,n_k) with the property that ν(\Ix) >= v(s;n_1,...,n_k) always holds for s points in generic position in P^{n_1} x ... x P^{n_k}. When k=1, v(s;n_1) equals the expected value for ν(I_X) as predicted by the Ideal Generation Conjecture. If k >= 2, we show that there are cases with ν(\Ix) > v(s;n_1,...,n_k). However, computational evidence suggests that in many cases ν(\Ix) = v(s;n_1,...,n_k).
Over a field of characteristic $0$, we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of $n \times n$ matrices. This modifies a conjecture of Weyman and provides a complete answer to it.
Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic $p$, we show that a similar result holds in equicharacteristic $0$ under the additional hypothesis that the symbolic Rees algebra of $I$ is noetherian.
We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.
We study the defining equations of projective embeddings of the blowup of P^2 at a set of {d+1 \choose 2} number of points in generic position. To do this, we first generalize the notion of a matrix, its ideal of 2x2 minors to that of a box-shaped matrix. Our work completes previous works of Geramita and Gimigliano.
We describe a minimal set of generators of the defining ideal of the Rees Algebra associated to a proper parametrization of any monoid hypersurface. In the case of plane curves, we recover a known description for rational parametrizations having a syzygy of minimal degree (μ=1). We also show that our approach can be applied to parametrizations of rational surfaces having a Hilbert-Burch resolution with μ_1=μ_2=1.
If $I=(f_1,\ldots,f_r)$ is an ideal in $S=k[x_1,\ldots,x_n]$, and $f_i$ are "general" elements of given degrees, there is a conjecture on the Hilbert series of $S/I$. We are considering the corresponding concepts in bigraded rings.
Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.
The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety $V$ is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where $V$ is a reduced set of points in the projective plane. It is shown that the module is nonzero for several cases of a special configuration class therein -- called $(s-r)$-{fold collinear configuration}. A complete classification of types is given for $5$ and $6$ points in regard to this problem.
We investigate the properties of coordinate rings of orbit closures for quivers of type $A_3$ by considering the desingularization given by Reineke. We construct explicit minimal free resolutions of the defining ideals of the orbit closures thus giving us a minimal set of generators for the defining ideal. The resolution allows us to read off some geometric properties of the orbit closure. In addition, we give a characterization for the orbit closure to be Gorenstein.
Let X be a smooth projective variety with torsion-free Picard group. We introduce complexes of vector spaces whose homology determines the structure of the minimal free resolution of the Cox ring of X over the polynomial ring and show how the homology of these complexes can be studied by purely geometric methods. As an application of these techniques we give a simple new proof of a characterization of the Cox rings of Del Pezzo surfaces (of degree >1) conjectured by Batyrev and Popov.
This note computes a Gröbner basis for the ideal defining a union of Schubert varieties. More precisely, it computes a Gröbner basis for unions of schemes given by northwest rank conditions on the space of all matrices of a fixed size. Schemes given by northwest rank conditions include classical determinantal varieties and matrix Schubert varieties--closures of Schubert varieties lifted from the flag manifold to the space of matrices.
Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero dimensional subschemes of projective space; the methods and results obtained there have much less to say about higher dimensional subschemes. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the …
In this article we produce Groebner bases for the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers, correcting previous work of Sengupta,(2003).
The structure of the defining ideal of the semigroup ring $k[H]$ of a numerical semigroup $H$ over a field $k$ is described, when the pseudo-Frobenius numbers of $H$ are multiples of a fixed integer.
Let X be a set of smooth points in P^2, and I = \oplus_{t >= d} I_t the defining ideal of X. In this paper, we give a set of defining equations for the Rees algebra R(I_{d+1}) of the ideal generated by I_{d+1}. This study give information to completely answer questions on the defining ideals of projective embeddings of the blowup of P^2 along the points in X, which has been studied by many authors. In this paper, we also study the asymptotic behaviour of the Rees algebras R(I_t) of the ideal generated by I_t, as t gets large.
The purpose of this paper is to prove that the symbolic Rees rings of ideals defining certain finite sets of points in the projective plane over an algebraically closed field are finitely generated using a ring theoretical criterion which is known as Huneke's criterion.
We exhibit a set of minimal generators of the defining ideal of the Rees Algebra associated to the ideal of three bivariate homogeneous polynomials parametrizing a proper rational curve in projective plane, having a minimal syzygy of degree 2.
We prove that the defining ideal of a sufficiently high Veronese subring of a toric algebra admits a quadratic Gröbner basis consisting of binomials. More generally, we prove that the defining ideal of a sufficiently high Veronese subring of a standard graded ring admits a quadratic Gröbner basis. We give a lower bound on $d$ such that the defining ideal of $d$-th Veronese subring admits a quadratic Gröbner basis. Eisenbud--Reeves--Totaro stated the same theorem without a proof with some lower bound on $d$. In many cases, our lower bound is less than Eisenbud--Reeves--Totaro's lower bound.
We discuss projective equivalence of ideals in Noetherian rings and the existence or failure of existence of projectively full ideals. We describe connections with the Rees valuations and Rees integers of an ideal, and consider the question of whether improvements can be made by passing to an integral extension ring.
The set of common roots of a finite set $I$ (it is an ideal) of homogeneous polynomials is known as projective algebraic set $V$. In this article I show how to dualize such projective algebraic sets $V$ by elimination of variables from a system of polynomials with the Gröbner bases method. A dualization algorithm is implemented in the computer algebra system {\sc Singular}. Some examples are given. The main diagram shows the relationship between the ideal $I$, its radical $\sqrt{I}$ and their dual ideals.
In this paper we study the asymptotic behavior of the regularity of symbolic powers of ideals of points in a weighted projective plane. By a result of Cutkosky, Ein and Lazarsfeld, regularity of such powers behaves asymptotically like a linear function. We study the difference between regularity of such powers and this linear function. Under some conditions, we prove that this difference is bounded, or eventually periodic. As a corollary we show that, if there exists a negative curve, then the regularity of symbolic powers of a monomial space curve is eventually a periodic linear function. We …
In this paper, we study arithmetic Macaulayfication of projective schemes and Rees algebras of ideals. We give a necessary and sufficient condition for a nonsingular projective scheme to have an arithmetic Macaulayfication. If the scheme is not necessarily nonsingular, we show that our condition give a sufficient condition for the existence of an arithmetic Macaulayfication. We also study truncated Rees algebra of an ideal. We show that the truncated Rees algebra R_λ(I) of an ideal I is alway Cohen-Macaulay for λlarge enough in some important situations.
We define weighted projective Reed-Muller codes over a subset of weighted projective space over a finite field. We focus on the case when the set X is a projective weighted torus. We show that the vanishing ideal of X is a lattice ideal and relate it with the lattice ideal of a minimal presentation of the semigroup algebra of Q, the numerical semigroup generated by the weights of the projective space. We compute the index of regularity of the vanishing ideal as function of the weights and the Frobenius number of Q. We compute the basic parameters of weighted projective Reed-Muller codes over a…
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…
In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to vertex weighted rooted forests and oriented cycles. As some consequences, we give some exact formulas for the depth of these ideals.
Let S be a polynomial ring in n variables, over an arbitrary field. We give the total, graded, and multigraded Betti numbers of S/M, for every monomial ideal M in S. We also give an explicit characterization of all monomial ideals M in S for which the quotient S/M has projective dimension n.
We provide bounds on the Castelnuovo-Mumford regularity in terms of ``defining equations'' by using elements that annihilates some cohomology modules, inspired by works of Miyazaki, Nagel, Schenzel and Vogel. The elements in these annihilators are provided either by liaison or by tight closure theories. Our results hold in any characteristic.
The diagonal in a product of projective spaces is cut out by the ideal of 2x2-minors of a matrix of unknowns. The multigraded Hilbert scheme which classifies its degenerations has a unique Borel-fixed ideal. This Hilbert scheme is generally reducible, and its main component is a compactification of PGL(d)^n/PGL(d). For n=2 we recover the manifold of complete collineations. For projective lines we obtain a space of trees that is irreducible but singular. All ideals in our Hilbert scheme are radical. We also explore connections to affine buildings and Deligne schemes.
Let $R$ be a commutative ring with the unit element. It is shown that an ideal $I$ in $R$ is pure if and only if Ann$(f)+I=R$ for all $f\in I$. If $J$ is the trace of a projective $R$-module $M$, we prove that $J$ is generated by the ``coordinates" of $M$ and $JM = M$. These lead to a few new results and alternative proofs for some known results.
Given a standard graded polynomial ring over a commutative Noetherian ring $A$, we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when $A$ is a field of characteristic zero, and follows from a result of Peskine and Szpiro when $A$ is a field of positive characteristic; our result applies, for example, when $A$ is the ring of integers.
We show that given any polynomial ring R over a field, and any ideal J in R which is generated by three cubic forms, the projective dimension of R/J is at most 36. We also settle the question whether ideals generated by three cubic forms can have projective dimension greater than 4, by constructing one with projective dimension equal to 5.
We develop several techniques for the study of the radical ideal $I$ defining the diagonal locus of $(\C^2)^n$. Using these techniques, we give combinatorial construction of generators for $I$ of certain bi-degrees.
By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.
In this paper, we give a formula to compute all the top degree graded Betti numbers of the path ideals of a cycle. As a consequence we can give a formula to compute its projective dimension and regularity.
In this paper we provide some precise formulas for regularity of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs. For the projective dimension of such an edge ideal, we give its exact formula. Meanwhile, we also give the upper and lower bounds of projective dimension of higher power of such edge ideals. Some examples show that these formulas are related to direction selection.
In this short note we prove that the projective dimension of a sequentially Cohen-Macaulay square-free monomial ideal is equal to the maximal height of its minimal primes (also known as the big height), or equivalently, the maximal cardinality of a minimal vertex cover of its facet complex. This in particular gives a formula for the projective dimension of facet ideals of these classes of ideals, which are known to be sequentially Cohen-Macaulay: graph trees and simplicial trees, chordal graphs and some cycles, chordal clutters and graphs, and some path ideals to mention a few. Since polarizat…
We prove that certain modules are faithful. This enables us to draw consequences about the reduction number and the integral closure of some classes of ideals.
This article will appear in the proceedings of the AMS Summer Institute in Algebraic Geometry at Santa Cruz, July 1995. The topic is toric ideals, by which I mean the defining ideals of subvarieties of affine or projective space which are parametrized by monomials. Numerous open problems are given.
We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective degrees of rational maps determined by perfect ideals of height two or by Gorenstein ideals of height three.
We parametrize the affine space of Artinian affine ideals of K[x,y] which have a given initial ideal with respect to the degree reverse lexicographic term order. The fact that the term order is degree compatible allows us to extend the parametrization to the projective case, namely zero-dimensional subschemes of the projective plane, with some extra assumption and to determine the Betti strata of these Groebner cells. This allows us to prove a formula due to A. Iarrobino for the codimension of the Betti strata of codimension two punctual schemes in the projective plane.
We introduce an invariant, associated to a coherent sheaf over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and a*-invariant of powers of homogeneous ideals.
We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An i…
We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of m polynomials in n variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results.
We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the $S/I$. We also provide a new characterization of quasi-trees, which we use to give a new proof to a result by Herzog, Hibi, and Zheng which characterizes monomial ideals of projective dimension 1 in terms of quasi-trees.
We define a family of homogeneous ideals with large projective dimension and regularity relative to the number of generators and their common degree. This family subsumes and improves upon constructions given in [Cav04] and [McC]. In particular, we describe a family of three-generated homogeneous ideals in arbitrary characteristic whose projective dimension grows asymptotically as sqrt{d}^(sqrt(d) - 1).
For the ideal $\mathfrak{p}$ in $k[x, y, z]$ defining a space monomial curve, we show that $\mathfrak{p}^{(2 n - 1)} \subseteq \mathfrak{m} \mathfrak{p}^{n}$ for some positive integer $n$, where $\mathfrak{m}$ is the maximal ideal $(x, y, z)$. Moreover, the smallest such $n$ is determined. It turns out that there is a counterexample to a claim due to Grifo, Huneke, and Mukundan, which states that $\mathfrak{p}^{(3)} \subseteq \mathfrak{m} \mathfrak{p}^2$ if $k$ is a field of characteristic not $3$; however, the stable Harbourne conjecture holds for space monomial curves as they claimed.
This article first presents two examples of algorithms that extracts information on scheme out of its defining equations. We also give a review on the notion of Castelnuovo-Mumford regularity, its main properties (in particular its relation to computational issues) and different ways that were used to estimate it.
A generalization of the plane de Jonquières transformation to arbitrary dimension is studied, with an eye for the ideal theoretic side. In particular, one considers structural properties of the corresponding base ideal and of its defining relations. Useful throughout is the idea of downgraded sequences of forms, a tool considered in many sources for the rounding-up of ideals of defining relations. The emphasis here is on the case where the supporting Cremona transformation of the de Jonquières transformation is the identity map. In this case we establish aspects of the homological behavior of …
