Links of symbolic powers of prime ideals

Symbolic powers of certain cover ideals of graphs

In this paper, we compute the regularity and Hilbert series of symbolic powers of the cover ideal of a graph $G$ when $G$ is either a crown graph or a complete multipartite graph. We also compute the multiplicity of symbolic powers of cover ideals in terms of the number of edges.

单项式理想的符号亏缺

Symbolic Defect of Monomial Ideals

本文研究单项式理想的符号幂与通常幂的差异，提出两个量化函数并确定其渐近增长率，同时利用符号多面体进行显式计算。

三次循环图的边理想的幂与符号幂的正则性

Regularity of Powers and symbolic powers of edge ideals of cubic circulant graphs

本文计算了所有三次循环图的边理想的幂与符号幂的正则性，并证明了三次循环图情形下的Minh猜想。

On the Equality of Ordinary and Symbolic Powers of Ideals

We consider the following question concerning the equality of ordinary and symbolic powers of ideals. In a regular local ring, if the ordinary and symbolic powers of a one-dimensional prime ideal are the same up to its height, then are they the same for all powers? We provide supporting evidence of a positive answer for classes of prime ideals defining monomial curves or rings of low multiplicities.

Regularity of Symbolic Powers of Edge Ideals

In this article, we prove that for several classes of graphs, the Castelnuovo-Mumford regularity of symbolic powers of their edge ideals coincide with that of their ordinary powers.

Symbolic powers of determinantal ideals in prime characteristic

We study the symbolic powers of determinantal ideals of generic, generic symmetric, and Hankel matrices of variables, and of Pfaffians of generic skew-symmetric matrices, in prime characteristic. Specifically, we show that the limit $\lim\limits_{n\to\infty} \frac{\textrm{reg}(I^{(n)})}{n}$ exists and that $\textrm{depth}(R/I^{(n)})$ stabilizes for $n\gg 0$. Furthermore, we give explicit formulas for the stable value of $\textrm{depth}(R/I^{(n)})$ in the generic and skew-symmetric cases. In order to show these results, we introduce the notion of symbolic $F$-purity of ideals which is satisfied…

Finitely generated symbolic Rees rings of ideals defining certain finite sets of points in P^2

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.

Upper bounds for the regularity of symbolic powers of certain classes of edge ideals

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers.

Symbolic Rees algebras and set-theoretic complete intersections

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement.

Symbolic powers of perfect ideals of codimension 2 and birational maps

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…

Comparison of symbolic and ordinary powers of ideals

In this paper we generalize the theorem of Ein-Lazarsfeld-Smith (concerning the behavior of symbolic powers of prime ideals in regular rings finitely generated over a field of characteristic 0) to arbitrary regular rings containing a field. The basic theorem states that in such rings, if P is a prime ideal of height c, then for all n, the symbolic (cn)th power of P is contained in the nth power of P. Results are also given in the non-regular case: one must correct by a power of the Jacobian ideal in rings where the Jacobian ideal is defined.

Invariants of the symbolic powers of edge ideals

Let $G$ be a graph and $I=I(G)$ be its edge ideal. When $G$ is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of $I$ and compute the Waldschmidt constant. When $G$ is complete graph then we describe the generators of the symbolic powers of $I$ and compute the Waldschmidt constant and the resurgence of $I$. Moreover for complete graph we prove that the Castelnuvo-Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

Binomial expansion for saturated and symbolic powers of sums of ideals

There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordinary powers. We prove a binomial expansion formula for saturated powers of sums of ideals. This gives a uniform treatment to an array of existing and new results on both notions of symbolic powers of such sums: binomial expansion formulas, computations of depth and regularity, and …

Intersections of symbolic powers of prime ideals

Let (R,m) be a local ring with prime ideals p and q such that p+q is an m-primary ideal. If R is regular and contains a field, and dim(R/p)+dim(R/q)=dim(R), we prove that p^{(r)}\cap q^{(n)}\subseteq m^{m+n} for all positive integers r and s. This is proved using a generalization of Serre's Intersection Theorem which we apply to a hypersurface R/fR. The generalization gives conditions that guarantee that Serre's bound on the intersection dimension dim(R/p)+dim(R/q) \leq dim(R) holds when R is nonregular.

Expected resurgences and symbolic powers of ideals

We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves.

On symbolic powers of prime ideals

Let (R,m) be a regular local ring with prime ideals p and q such that p+q is m-primary and dim(R/p)+dim(R/q)=dim(R). It has been conjectured by Kurano and Roberts that p^{(n)} \cap q \subseteq m^{n+1} for all positive integers n. We discuss this conjecture and related conjectures. In particular, we prove that this conjecture holds for all regular local rings if and only if it holds for all localizations of polynomial algebras over complete discrete valuation rings. In addition, we give examples showing that certain generalizations to nonregular rings do not hold.

Symbolic Powers of Cover Ideals of Graphs and Koszul Property

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

极值理想与积分闭包 via 极端理想

Symbolic powers and integral closures via extremal ideals

本文通过极值理想计算平方自由单项式理想幂和符号幂的积分闭包，并提供精确的不变量界。

Symbolic defects of edge ideals of unicyclic graphs

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let $G$ be a unicyclic graph with a unique odd cycle and $I=I(G)$ be its edge ideal. We compute the exact values of all symbolic defects of $I$ using the concept of minimum edge cover for an induced subgraph in a graph. We describe one method to find the quasi-polynomial associated with the symbolic defects of edge ideal $I$. We classify the class of unicyclic graphs when some power of maximal ideal annihilates $ I^{(s)}/I^s $ for any fixed $ s $. Also for those class of graphs, we compute the Hilbert function o…

Symbolic Integration in Prime Characteristic

In this paper we study elementary extensions of differential fields in prime characteristic. In particular, we show that, in contrast to Liouville's result in characteristic zero, all elements of an elementary extension admit an antiderivative in some logarithmic extension.

Symbolic powers of monomial ideals which are generically complete intersections

We classify all unmixed monomial ideals I of codimension 2 which are generically a complete intersection and which have the property that the symbolic power algebra A(I) is standard graded. We give a lower bound for the highest degree of a generator of A(I) in the case that I is a modification of the vertex cover ideal of a bipartite graph, and show that this highest degree can be any given number. We furthermore give an upper bound for the highest degree of a generator of the integral closure of A(I) in the case that I is a monomial ideal which is generically a complete intersection.

The Symbolic Generic Initial System of Almost Linear Point Configurations in P2

Consider an ideal I in K[x,y,z] corresponding to a point configuration in P2 where all but one of the points lies on a single line. In this paper we study the symbolic generic initial system obtained by taking the reverse lexicographic generic initial ideals of the corresponding uniform fat point ideals. We describe the limiting shape of this system of ideals and, in proving this result, demonstrate that infinitely many of the uniform fat point ideals are componentwise linear.

The symbolic defect of an ideal

Let $I$ be a homogeneous ideal of $\Bbbk[x_0,\ldots,x_n]$. To compare $I^{(m)}$, the $m$-th symbolic power of $I$, with $I^m$, the regular $m$-th power, we introduce the $m$-th symbolic defect of $I$, denoted $\operatorname{sdefect}(I,m)$. Precisely, $\operatorname{sdefect}(I,m)$ is the minimal number of generators of the $R$-module $I^{(m)}/I^m$, or equivalently, the minimal number of generators one must add to $I^m$ to make $I^{(m)}$. In this paper, we take the first step towards understanding the symbolic defect by considering the case that $I$ is either the defining ideal of a star configu…

Packing properties of cubic squarefree monomial ideals

The symbolic powers, in general, are not equal to the ordinary powers. Therefore, one interesting question here is for what classes of ideals ordinary and symbolic powers coincide? The answer to this question for squarefree monomial ideals may be packing property. In this paper, we classify all cubic path ideals for those the symbolic and ordinary powers coincide.

Equality of ordinary and symbolic powers of Stanley-Reisner ideals

This paper studies properties of simplicial complexes for which the m-th symbolic power of the Stanley-Reisner ideal equals to the m-th ordinary power for a given m > 1. The main results are combinatorial characterizations of such complexes in the two-dimensional case. It turns out that there exist only a finite number of complexes with this property and that these complexes can be described completely. As a consequence we are able to determine all complexes for which the m-th ordinary power of the Stanley-Reisner ideal is Cohen-Macaulay for a given m > 1.

Regularity of quasi-symbolic and bracket powers of Borel type ideals

In this paper, we show that the regularity of the q-th quasi-symbolic power $I^{((q))}$ and the regularity of the $q$-th bracket power $I^{[q]}$ of a monomial ideal of Borel type $I$, satisfy the relations $reg(I^{((q))})\leq q \cdot reg(I)$, respectively $reg(I^{[q]})\geq q\cdot reg(I)$. Also, we give an upper bound for $reg(I^{[q]})$.

Regularity of symbolic powers of square-free monomial ideals

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_Δ$ is the Stanley-Reisner ideal of a simplicial complex $Δ$, then $\reg(I^{(n)}) \leqslant δ(n-1) +b$ for all $n\geqslant 1$, where $δ= \lim\limits_{n\to\infty} \reg(I^{(n)})/n$, and $b = \max\{\reg(I_Γ) \mid Γ\text{ is a subcomplex of } Δ\text{ with } \F(Γ) \subseteq \F(Δ)\}$. This bound is sharp for any $n$. When $I = I(G)$ is the edge ideal of a simple graph $G$, we obtain a general linear upper bound $\reg(I^{(n)}) \leqslant 2n + \ordmatch(G)-1$, where $\ordmatch(G)$ is the ordered matching n…

Ordinary and symbolic powers are Golod

Let $S$ be a positively graded polynomial ring over a field of characteristic 0, and $I\subset S$ a proper graded ideal. In this note it is shown that $S/I$ is Golod if $\partial(I)^2\subset I$. Here $\partial(I)$ denotes the ideal generated by all the partial derivatives of elements of $I$. We apply this result to find large classes of Golod ideals, including powers, symbolic powers, and saturations of ideals.

Comparision between regularity of small symbolic powers and ordinary powers of an edge ideal

Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${\rm reg}(I^{(s)}) = {\rm reg}(I^s)$$ for $s = 2,3$, where $I^{(s)}$ is the $s$-th symbolic power of $I$. As a consequence, we prove the following bounds \begin{align*} {\rm reg} I^{s} & \le {\rm reg} I + 2s - 2, \text{ for } s = 2,3, {\rm reg} I^{(s)} & \le {\rm reg} I + 2s - 2,\text{ for } s = 2,3,4. \end{align*}

Symbolic powers of ideals defining F-pure and strongly F-regular rings

Given a radical ideal $I$ in a regular ring $R$, the Containment Problem of symbolic and ordinary powers of $I$ consists of determining when the containment $I^{(a)} \subseteq I^b$ holds. By work of Ein-Lazersfeld-Smith, Hochster-Huneke and Ma-Schwede, there is a uniform answer to this question, but the resulting containments are not necessarily best possible. We show that a conjecture of Harbourne holds when $R/I$ is F-pure, and prove tighter containments in the case when $R/I$ is strongly F-regular.

Comparing symbolic powers of edge ideals of weighted oriented graphs

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. If $D$ contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of a weighted oriented graph having each edge in some induced odd cycle of it. We characterize the weighted naturally oriented unicyclic graphs with unique odd cycles and weighted naturally oriented even cycles for the equality of ordinary and symbolic powers of their edge ideals. Let $ D^{\prime}…

The asymptotic behaviour of symbolic generic initial systems of points in general position

Consider the ideal I corresponding to r points in P^2. We study the symbolic generic initial system of I, formed by taking the generic initial ideals of the symbolic powers of I, and its asymptotic behaviour. In particular, we describe the limiting shape of this system explicitly when the points lie in general position using the SHGH Conjecture for more than eight points. The symbolic generic initial system and its limiting shape reflects information about the Hilbert functions of uniform fat point ideals.

图的覆盖理想的符号幂的V-数

V-numbers of symbolic powers of cover ideals of graphs

该研究针对图的覆盖理想的符号幂，证明了二分图与非二分图对应符号幂的局部v-数的不同变化规律，并给出了基于图组合数据的显式不变量公式。

On the prime ideal structure of symbolic Rees algebras

This paper contributes to the study of the prime spectrum and dimension theory of symbolic Rees algebra over Noetherian domains. We first establish some general results on the prime ideal structure of subalgebras of affine domains, which actually arise, in the Noetherian context, as domains between a domain $A$ and $A[a^{-1}]$. We then examine closely the special context of symbolic Rees algebras (which yielded the first counter-example to the Zariski-Hilbert problem). One of the results states that if $A$ is a Noetherian domain and $p$ a maximal ideal of $A$, then the Rees algebra of $p$ inhe…

Symbolic powers of edge ideals of graphs

Let $G$ be a graph and let $I = I(G)$ be its edge ideal. When $G$ is unicyclic, we give a decomposition of symbolic powers of $I$ in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the resurgence number of $I$. When $G$ is an odd cycle, we explicitly compute the regularity of $I^{(s)}$ for all $s \in \mathbb{N}$. In doing so, we also give a natural lower bound for the regularity function $\text{reg } I^{(s)}$, for $s \in \mathbb{N}$, for an arbitrary graph $G$.

Symbolic powers of multipartite hypergraphs and Waldschmidt constant

In this paper, we find a criterion to answer when the second symbolic and ordinary powers of the edge ideal of a 3-partite hypergraph are equal. Also, we give the formula to compute the Waldschmidt constant of the path ideals of a cycle.

Symbolic powers of ideals

We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject.

可接受子图与图的覆盖理想符号幂的深度

Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs

本文引入了t-可接受子图的概念，并利用其计算图的覆盖理想t次符号幂的深度，证明了对于循环图C_n，其符号幂的深度公式。

Regularity and Koszul property of symbolic powers of monomial ideals

Let $I$ be a homogeneous ideal in a polynomial ring over a field. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. Motivated by results about ordinary powers of $I$, we study the asymptotic behavior of the regularity function $\text{reg}~ (I^{(n)})$ and the maximal generating degree function $ω(I^{(n)})$, when $I$ is a monomial ideal. It is known that both functions are eventually quasi-linear. We show that, in addition, the sequences $\{\text{reg}~ I^{(n)}/n\}_n$ and $\{ω(I^{(n)})/n\}_n$ converge to the same limit, which can be described combinatorially. We construct an example of an equidi…

Cohen-Macaulayness of large powers of Stanley-Reisner ideals

We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex is a matroid. Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only if the complex is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.

Symbolic powers of ideals and their topology over a module

Let $I$ denote an ideal of a Noetherian ring $R$ and $N$ a non-zero finitely generated $R$-module. In the present paper, some necessary and sufficient conditions are given to determine when the $I$-adic topology on $N$ is equivalent to the $I$-symbolic topology on $N$. Among other things, we shall give a complete solution to the question raised by R. Hartshorne in [{\it Affine duality and cofiniteness}, Invent. Math. {\bf9}(1970), 145-164], for a prime ideal $\frak p$ of dimension one in a local Noetherian ring $R$, by showing that the $\frak{p}$-adic topology on $N$ is equivalent to the $\fra…

理想滤过的$F$-阈值

$F$-thresholds of filtrations of ideals

该研究将理想的$F$-阈值推广到理想滤过情形，证明多类滤过$F$-阈值存在性，给出有限性充要条件、符号$F$-阈值上界与符号$F$-分裂充分条件，并计算了多类理想的符号$F$-阈值。

Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers. We give a systematic treatment of the family of Fermat ideals, describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers. We use these to study the ordinary and the symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbol…

On the stable set of associated prime ideals of a monomial ideal

It is shown that any set of nonzero monomial prime ideals can be realized as the stable set of associated prime ideals of a monomial ideal. Moreover, an algorithm is given to compute the stable set of associated prime ideals of a monomial ideal.

Toric理想的符号幂

Symbolic Powers of Toric Ideals

本文研究toric理想的符号幂，利用格结构导出的线性映射的核加以刻画，并证明其可表示为普通幂关于全体变量乘积单项式的饱和，给出具有计算价值的结果。

Symbolic powers of codimension two Cohen-Macaulay ideals

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…

无平方单项式理想的符号幂与普通幂的v数比较

Comparing v-numbers of symbolic and ordinary powers of squarefree monomial ideals

该研究针对无平方单项式理想，建立了其普通幂与符号幂的v数的上界，指出二者差值可任意大，并确定了路和圈的边理想对应幂的v数。

Borel Fixed Initial Ideals of Prime Ideals in Dimension Two

We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated prime ideal of dimension one larger.

The stable set of associated prime ideals of a squarefree principal Borel ideal

It is shown that a squarefree principal Borel ideal satisfies the persistence property for the associated prime ideals. For the graded maximal ideal we compute the index of stability with respect to squarefree principal Borel ideals and determine their stable set of associated prime ideals.

Symbolic powers and free resolutions of generalized star configurations of hypersurfaces

As a generalization of the ideals of star configurations of hypersurfaces, we consider the $a$-fold product ideal $I_a(f_1^{m_1}\cdots f_s^{m_s})$ when ${f_1,\dots,f_s}$ is a sequence of generic forms and $1\le a\le m_1+\cdots+m_s$. Firstly, we show that this ideal has complete intersection quotients when these forms are of the same degree and essentially linear. Then we study its symbolic powers while focusing on the uniform case with $m_1=\cdots=m_s$. For large $a$, we describe its resurgence and symbolic defect. And for general $a$, we also investigate the corresponding invariants for meeti…

空间单项素理想度数5的符号Rees代数

Symbolic Rees algebras of space monomial primes of degree 5

本文研究了空间单项素理想度数5的符号Rees代数，证明其不是Noetherian的。

关于二项边理想的符号$F$-分裂性

On the symbolic $F$-splitness of binomial edge ideals

本文研究了二项边理想族的符号$F$-分裂性及其符号吹胀代数的强$F$-正则性，利用Fedder类准则和相关图的组合性质对上述情形进行了探讨。

Generalized test ideals and symbolic powers

Hochster and Huneke proved in \cite{HH5} fine behaviors of symbolic powers of ideals in regular rings, using the theory of tight closure. In this paper, we use generalized test ideals, which are a characteristic $p$ analogue of multiplier ideals, to give a slight generalization of Hochster-Huneke's results.

与棋盘复形相关的理想的幂的 $\mathrm{v}$-数

The $\mathrm{v}$-number of powers of ideals associated with chessboard complexes

本文研究棋盘复形的面理想和Stanley--Reisner理想的普通幂与符号幂的$\mathrm{v}$-数，建立了精确公式，并证明普通幂的$\mathrm{v}$-数函数渐近线性。