Leah H. Gold
Herzog and Srinivasan have conjectured that for any homogeneous k-algebra, the degree is bounded above by a function of the maximal degrees of the syzygies. Combining the syzygy quadrangle decomposition of Peeva and Sturmfels and a delicate case analysis, we prove that this conjectured bound holds for codimension 2 lattice ideals.
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Structure theorems for Gorenstein ideals of codimension four with small number of generators
本文用表示论方法研究余维数为四的Gorenstein理想的极小自由消解，引入高阶结构映射族，并证明由六个元素极小生成的该类理想必为余维数为三的Gorenstein理想的超平面截面，同时提出七、八个生成元情形的猜想。
In this note, we calculate the Koszul homology of the codimension 3 Gorenstein ideals. We find filtrations for the Koszul homology in terms of modules with pure free resolutions and completely describe these resolutions. We also consider the Huneke-Ulrich deviation 2 ideals.
We give a structure theorem for Cohen Macaulay monomial ideals of codimension 2, and describe all possible relation matrices of such ideals. In case that the ideal has a linear resolution, the relation matrices can be identified with the spanning trees of a connected chordal graph with the property that each distinct pair of maximal cliques of the graph has at most one vertex in common.
In this paper we define an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the Tor algebra. This family is likely to play a key role in classifying perfect ideals with five generators of type two.
Let K be a field and S = K[x1,...,xn] be a polynomial ring. A single spot ideal I =< S is a graded ideal whose local cohomology H^i_\mm(S/I), i< dim S/I and \mm = (x1,...,xn), only has non-trivial value N, a finite length module, at i = depth S/I. We consider characterization of single spot ideals in terms of (long) Bourbaki sequences. The codimension 2 case has been fairly well investigated. In this paper, we focus on the codimension 3 case.
We study Gorenstein ideals of codimension $4$ derived from generic doublings of almost complete intersection perfect ideals of codimension $3$. We also investigate spinor coordinates of such Gorenstein ideals with $8$ and $9$ generators. For an ideal $J$ of commutative ring $R$, the $R/J$ module $J/J^2$ is called conormal module and $R/J$-dual of $J/J^2$ is called normal module. We study properties of conormal and normal modules of almost complete intersection perfect ideals of codimension $3$.
One deals with catalectic codimension two perfect ideals and certain degenerations thereof, with a view towards the nature of their symbolic powers. In the spirit of [10] one considers linearly presented such ideals, only now in the situation where the number of variables is sufficiently larger than the size of the matrix, yet still stays within reasonable bounds.
This work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field, where the entries of the corresponding presentation matrix are general linear forms. The main contribution of the present approach is the use of the birational theory underlying the nature of the ideal and the details of a deep interlacing between generators of its symbolic powers and the inversion factors stemming from the inverse map to the birational map defined by the linear system spanned by the generators of this ideal. A full description of the corresponding symbolic Rees alg…
We analyze the structure of spinor coordinates on resolutions of Gorenstein ideals of codimension four. As an application we produce a family of such ideals with seven generators which are not specializations of Kustin-Miller model.
We show that the union of certain codimension two flats in Fermat hyperplane arrangements does not have the symbolic cube of the defining ideal contained in the square of the ideal.
Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective …
Every normal toric ideal of codimension two is minimally generated by a Grobner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of fixed codimension is normal.
A homogeneous ideal is robust if its universal Gröbner basis is also a minimal generating set. For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable binomials. We characterize the codimension 2 strongly robust toric ideals by their Gale diagrams. This gives a positive answer to a question of Petrovic, Thoma, and Vladoiu in the case of codimension 2 toric ideals.
Generic linkage is used to compute a prime ideal such that the radical of the initial ideal of the prime ideal is equal to the radical of a given codimension two monomial ideal that has a Cohen-Macaulay quotient ring.
In this paper, we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using characteristic $p$ methods. When $X$ is a normal Gorenstein closed subvariety of a smooth complex variety $A$, we formulate a restriction property of the adjoint ideal sheaf $\adj_X(A)$ of $A$ along $X$ involving the l.c.i. ideal sheaf $\mathcal{D}_X$ of $X$. The proof relies on a modification of generalized test ideals of Hara and Yoshida.
We provide explicit combinatorial descriptions of the primary components of codimension two lattice basis ideals. As an application, we compute the set of parameters for which a bivariate Horn system of hypergeometric differential equations is holonomic.
Let $I_X$ be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme $X \subseteq \mathbb{P}^n$, and let $I_X^{(m)}$ denote its $m$-th symbolic power. We are interested in when $I_X^{(m)} = I_X^m$. We survey what is known about this problem when $X$ is locally a complete intersection, and in particular, we review the classification of when $I_X^{(m)} = I_X^m$ for all $m \geq 1$. We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to…
We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this degree comes along with the minimal number of generators of $I$ and the degree of the entries of the associated skew-symmetric matrix in a simple formula. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal $I$ of codimension three filling them. We conjecture that, f…
Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of monomial ideals. Using this we classify when both bounds are sharp. Moreover, we prove that the artinian quotient of a monomial ideal in $K[x,y]$ always has the strong Lefschetz property, regardless of the characteristic of the field, exactly when the ideal is lexsegment. As a consequence we describe a family of non-monomial complete intersections that alwa…
It is unknown if an Artinian level O-sequence of codimension 3 and type $r (\ge 2)$ is unimodal, while it is known that any Gorenstein O-sequence of codimension 3 is unimodal. We show that some Artinian non-unimodal O-sequence of codimension 3 cannot be level. We also find another non-level case: if some Artinian algebra $A$ of codimension 3 has the Hilbert function $$\begin{matrix}\H & : & h_0 & h_1 & ... & h_{d-1} & \underbrace{h_d ... h_d}_{s\text{-times}} & h_{d+s}, \end{matrix} $$ such that $h_d<h_{d+s}$ and $s\ge 2$, then $A$ has a socle element in degree $d+s-2$, that is, $A$ is not lev…
We prove a sharp upper bound for the projective dimension of ideals of height two generated by quadrics in a polynomial ring with arbitrary large number of variables.
The main goal of this paper is to characterize the Hilbert functions of all (artinian) codimension 4 Gorenstein algebras that have at least two independent relations of degree four. This includes all codimension 4 Gorenstein algebras whose initial relation is of degree at most 3. Our result shows that those Hilbert functions are exactly the so-called {\em SI-sequences} starting with (1,4,h_2,h_3,...), where $h_4 \leq 33$. In particular, these Hilbert functions are all unimodal. We also establish a more general unimodality result, which relies on the values of the Hilbert function not being too…
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.
Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous Gorenstein ideal of codimension $g\ge3$ and initial degree $p\ge2$. We prove that the number of minimal generators $ν(I_p)$ of $I$ that are in degree $p$ is bounded above by $ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$, which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension $g$ and initial degree $p$. Further, $I$ is itself extremal if $ν(I_p)=ν_0$.
The famous structure theorem of Buchsbaum and Eisenbud gives a complete characterization of Gorenstein ideals of codimension 3 and their minimal free resolutions. We generalize the ideas of Buchsbaum and Eisenbud from Gorenstein ideals to Gorenstein algebras and present a description of Gorenstein algebras of any odd codimension. As an application we study the canonical ring of a numerical Godeaux surface.
We study Stanley decompositions and show that Stanley's conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3.
Let $(S, \mathfrak n) $ be a regular local ring and let $I \subseteq \mathfrak n^2 $ be a perfect ideal of $S. $ Sharp upper bounds on the minimal number of generators of $I$ are known in terms of the Hilbert function of $R=S/I. $ Starting from information on the ideal $I, $ for instance the minimal number of generators, a difficult task is to find good bounds on the minimal number of generators of the leading ideal $I^* $, which defines the tangent cone of $R$ or to give information on its graded structure. Motivated by papers of S.C. Kothari, S. Goto ${\it{et al.}}$ concerning the leading id…
The structure of almost Cohen-Macaulay $3$-generated ideals of codimension $2$ in terms of matrix theory
研究通过矩阵理论描述了codimension为2的几乎Cohen-Macaulay 3生成理想结构，基于自由分解和潜在数据，引入了水平矩阵概念，并给出其存在条件。
Cohen--Macaulay ideals of codimension two and the geometry of plane points
本文通过分析 Hilbert–Burch 矩阵研究余维数二 Cohen–Macaulay 理想及平面点几何，并证明超平面排列 Jacobian 理想猜想。
In 2018 Celikbas, Laxmi, Kraśkiewicz, and Weyman exhibited an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the Tor algebra. All previously known perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two had been found by Brown in 1987. Brown's ideals all have non-trivial multiplication on the Tor algebra. We prove that all of the ideals of Brown are obtained from the ideals of Celikbas, Laxmi, Kraśkiewicz, and Weyman by (non-homogeneous) specialization. We also prove that b…
The goal of this paper is the fine structure of the ideals in the title, with emphasis on the properties of the associated Rees algebra and the special fiber. The watershed between the present approach and some of the previous work in the literature is that here one does not assume that the ideals in question satisfy the common generic properties. One exception is a recent work of N. P. H. Lan which inspired the present work. Here we recover and extend his work. We strongly focus on the behavior of the ideals of minors of the corresponding so-called Hilbert--Burch matrix and on conjugation fea…
We study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Apéry set of a numerical semigroup generated by four natural numbers. We show that these algebras have the weak Lefschetz property whenever the initial degree of their defining ideal is small.
We consider a classification problem of ideals by codimension in case rings are the local rings of irreducible curve singularities. In this paper, we introduce a systematic method to solve this problem.
In the present note we study certain arrangements of codimension $2$ flats in projective spaces, we call them "Fermat arrangements". We describe algebraic properties of their defining ideals. In particular, we show that they provide counterexamples to an expected containment relation between ordinary and symbolic powers of homogeneous ideals.
We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences of these total degrees, p and q, are relatively prime, the extremal rays are parametrised by order ideals in N^2 contained in the region px + qy < (p-1)(q-1). We also consider some examples concerning artinian modules of codimension three.
We begin the study of the notion of diameter of an ideal I of a polynomial ring S over a field, an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals with diameter not larger than the codimension, such as: quadratic radical ideals of codimension at most 4 and such that S/I is Gorenstein, or ideals admitting a square-free complete intersection initial ideal.
We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determine…
The class of equidimensional polymatroidal ideals are studied. In particular, we show that an unmixed polymatroidal ideal is connected in codimension one if and only if it is Cohen-Macaulay. Especially a matroidal ideal is connected in codimension one precisely when it is a squarefree Veronese ideal. As a consequence we indicate that for polymatroidal ideals, the Serre's condition $(S_n)$ for some $n\geq 2$ is equivalent to Cohen-Macaulay property. We also give a classification of generalized Cohen-Macaulay polymatroidal ideals.
We find a sufficient condition that $\H$ is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function $\H=(h_0,h_1,..., h_{d-1}>h_d=h_{d+1})$ cannot be level if $h_d\le 2d+3$, and that there exists a level O-sequence of codimension 3 of type $\H$ for $h_d \ge 2d+k$ for $k\ge 4$. Furthermore, we show that $\H$ is not level if $β_{1,d+2}(I^{\rm lex})=β_{2,d+2}(I^{\rm lex})$, and also prove that any codimension 3 Artinian graded algebra $A=R/I$ cannot be level if $β_{1,d+2}(\Gin(I))=β_{2,d+2}(\Gin(I))$. In this case, the H…
I describe the projective resolution of a codimension 4 Gorenstein ideal, aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The main result is a structure theorem stating that the ideal is determined by its (k+1) x 2k matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety SpH_k in Mat(k+1,2k). This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.
Jordan degree type for codimension three Gorenstein algebras of small Sperner number
该研究将余维数2分次代数的Jordan次数型研究拓展至小Sperner数的余维数3分次Artinian Gorenstein代数，完全确定了s=3,4,5时对应Gorenstein序列的所有可能JDT，并划定了s=6时的潜在JDT范围。
Let X be a complete toric variety with homogeneous coordinate ring S. In this article, we compute upper and lower bounds for the codimension in the critical degree of ideals of S generated by dim(X)+1 homogeneous polynomials that don't vanish simultaneously on X.
We define the monomial invariants of a projective variety $Z$; they are invariants coming from the generic initial ideal of $Z$. Using this notion, we generalize a result of Cook: If $Z$ is an integral variety of codimension two, satisfying the additional hypothesis $s_Z=s_Γ,$ then its monomial invariants are connected.
We study the Rees algebra of a perfect Gorenstein ideal of codimension 3 in a hypersurface ring. We provide a minimal generating set of the defining ideal of these rings by introducing a modified Jacobian dual and applying a recursive algorithm. Once the defining equations are known, we explore properties of these Rees algebras such as Cohen-Macaulayness and Castelnuovo-Mumford regularity.
In this paper, we continue the study of which $h$-vectors $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$ can be the Hilbert function of a level algebra by investigating Artinian level algebras of codimension 3 with the condition $β_{2,d+2}(I^{\rm lex})=β_{1,d+1}(I^{\rm lex})$, where $I^{\rm lex}$ is the lex-segment ideal associated with an ideal $I$. Our approach is to adopt an homological method called {\it Cancellation Principle}: the minimal free resolution of $I$ is obtained from that of $I^{\rm lex}$ by canceling some adjacent terms of the same shift. We prove that when $β_{1,d+2}(I^{\rm lex})=β_{…
A note on Jordan types and Jordan degree types
本文探讨了 Jordan 次数类型是否比线性形式的 Jordan 类型包含更多分次 Artinian Gorenstein 代数的信息，证明在余维数二中前者由后者决定，但在高余维数中则不然。
The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture for the case of codimension three graded Gorenstein algebras.
This article is built upon three main ideas. First, for a class of monomial ideals, it is proven that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for an arbitrary graph G, we construct a monomial ideal M_G, and show that the chromatic number of G is equal to the codimension of M_G. Finally, for a class of graphs, we give a formula that computes the chromatic polynomial of G, evaluated at the chromatic number of G, in terms of the codimension and multiplicity of M_G. In particular, the formula applies to al…
Generic models of licci ideals parametrized by Schur functors
本文利用与 Schur 函子相关的分区对参数化余维 3 的 licci 理想的 Herzog 类，并组合描述其链接图，进而得到 Tor 代数乘法分类及结构定理。
The acquisition of the defining equations of Rees algebras is a natural way to study these algebras and allows certain invariants and properties to be deduced. In this paper, we consider Rees algebras of codimension 2 perfect ideals of hypersurface rings and produce a minimal generating set for their defining ideals. Then, using generic Bourbaki ideals, we study Rees algebras of modules with projective dimension one over hypersurface rings. We describe the defining ideal of such algebras and determine Cohen-Macaulayness and other invariants.
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.
In this paper we study squarefree monomial ideals which have constant depth functions. Edge ideals, matroidal ideals and facet ideals of pure simplicial forests connected in codimension one with this property are classified.
We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree $i+1$ entry of a Gorenstein $h$-vector, in terms of its entry in degree $i$. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given $r$ and $i$, all Gorenstein $h$-vectors of codimension $r$ and socle degree $e\geq e_0=e_0(r,i)$ (this function being explicitly computed) are unimodal up to degree $i+1$. This immediately gives a new proof of a theorem of Stanle…
We determine the codimension of the Betti strata of the family G(H) parametrizing graded Artinian quotients A of the polynomial ring R in two variables, having Hilbert function H. The Betti stratum G(B,H) parametrizes all such quotients having the graded Betti numbers determined by the relation degrees B. Our method is to identify G(B,H) as the product of determinantal varieties. We then use a result of M. Boij that reduces the calculation of codimension to showing that the most special stratum has the expected codimension. We also show that the closure of the Betti stratum is Cohen-Macaulay, …
The aim of the present exposition is to investigate varieties of almost minimal degree and of low codimension, in particular their Betti diagrams. Here minimal degree is defined as $°X = \codim X + 2.$ We describe the structure of the minimal free resolution of a variety $X$ of almost minimal degree of $\codim X \leq 4$ by listing the possible Betti diagrams. The most surprising fact is, that the non-arithmetically Cohen-Macaulay case of varieties of almost minimal degree can occur only in small dimensions (cf. Section 2 for the precise statements). Our main technical tool is a result shown by…
Some results on the Cohen-Macaulayness of the canonical module. We study the $S_2$-fication of rings which are quotients by lattices ideals. Given a simplicial lattice ideal of codimension two $I,$ its Macaulayfication is given explicitly from a system of generators of $I$.
We study how to lift Markov bases and Gröbner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the toric fiber product of toric ideals, where lifting gives Markov bases of the factor ideals that satisfy the compatible projection property. We illustrate the technique by computing Markov bases of various infinite families of hierarchical models. The methodology also implies new finiteness results for iterated toric fiber products.
Given a reduced local algebra $T$ over a suitable ring or field $k$ we study the question of whether there are nontrivial algebra surjections $R\to T$ which induce isomorphisms $Ω_{R/k}\otimes T \to Ω_{T/k}$. Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions of an algebra $T$ can be characterized in terms of the symbolic square of an ideal defining $T$. We give a characterization of the symbolic square in terms of Fitting ideals. Using thi…
We describe all Mathieu-Zhao spaces of $k[x_1,\cdots,x_n]$ ($k$ is an algebraically closed field of characteristic zero) which contains an ideal of finite codimension. Furthermore we give an algorithm to decide if a subspace of the form $I+kv_1+\cdots+kv_r$ is a Mathieu-Zhao space, in case the ideal $I$ has finite codimension.
A projectively normal Calabi-Yau threefold $X \subseteq \mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $\math…
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