Catalin Ciuperca, Florian Enescu, Sandra Spiroff
Let A be a locally analytically unramified local ring and let J_1,...,J_k,I be ideals in A. If C=C(J_1,...,J_k;I) is the cone generated by the (k+1)-tuples (m_1,...,m_k,n) such that J_1^{m_1}...J_k^{m_k} is contained in I^n, we prove that the topological closure of C is a rational polyhedral cone. This generalizes results by Samuel, Nagata and Rees.
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It was previously known, by work of Smith-Swanson and of Sharp-Nossem, that the linear growth property of primary decompositions of Frobenius powers of ideals in rings of prime characteristic has strong connections to the localization problem in tight closure theory. The localization problem has recently been settled in the negative by Brenner and Monsky, but the linear growth question is still open. We study growth of primary decompositions of Frobenius powers of dimension one homogeneous ideals in graded rings over fields. If the ring is positively graded we prove that the linear growth prop…
We study generalized symbolic powers and form ideals of powers of ideals and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.
Let $I$ be an ideal whose symbolic Rees algebra is Noetherian. For $m \geq 1$, the $m$-th symbolic defect, sdefect$(I,m)$, of $I$ is defined to be the minimal number of generators of the module $\frac{I^{(m)}}{I^m}$. We prove that sdefect$(I,m)$ is eventually quasi-polynomial as a function in $m$. We compute the symbolic defect explicitly for certain monomial ideals arising from graphs, termed cover ideals. We go on to give a formula for the Waldschmidt constant, an asymptotic invariant measuring the growth of the degrees of generators of symbolic powers, for ideals whose symbolic Rees algebra…
Asymptotic Resurgence of Facet and Stanley-Reisner ideals of Matroids
研究拟阵的斯坦利 - 赖斯纳理想的渐近复苏，通过与面理想渐近复苏的等价性，利用拟阵收缩的面理想的瓦尔施密特常数证明公式，得出渐近复苏与弱序的关系及相关上界，还证明了许多类拟阵面理想渐近复苏的下界等式并计算部分结果。
Let $R$ be a commutative Noetherian ring, $E$ a non-zero finitely generated $R$-module and $I$ an ideal of $R$. One purpose of this paper is to show that the sequences $\Ass_RE/ \widetilde{I_E^n}$ and $\Ass_R\widetilde{I^n _E}/\widetilde{I^{n+1}_E}$, $n = 1,2, \dots$, of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff \cite[Theorem 3.1]{MR}. In addition, a characterization concerning the set $\widetilde{A^*}(I,E)$ is included. A second purpose of this paper is to prove that $I$ has linear growth primary decompositions for …
We study chains of nonzero edge ideals that are invariant under the action of the monoid $\mathrm{Inc}$ of increasing functions on the positive integers. We prove that the sequence of Castelnuovo--Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behaviour sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of $\mathrm{Inc}$-invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of $\mathrm{Inc}$-invariant chains of …
This dissertation focuses on the following topics: (1) asymptotic prime divisors over complete intersection rings, (2) asymptotic stability of complexities over complete intersection rings, (3) asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules, and (4) characterizations of regular local rings via syzygy modules of the residue field.
Asymptotic Resurgence of Facet ideals of Graphic Matroids
该研究给出图拟阵的面理想的渐近复苏的上下界，二者对哈密顿图重合，且对顶点数≤9的简单2-连通非哈密顿图，其渐近复苏取值为上述上界或下界。
Asymptotic colengths for families of ideals: an analytic approach
本文研究Noetherian局部环中理想族渐近余长度的存在性，推广了多种理想族，并提出统一的解析方法证明极限存在，同时建立了相关不等式与体积=重数公式。
This paper is concerned with tight closure in a commutative Noetherian ring $R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal $I$ of $R$ has linear growth of primary decompositions, then tight closure (of $I$) commutes with localization at the powers of a single element. It is shown in this paper that, provided $R$ has a weak test element, linear growth of primary decompositions for other sequences of ideals of $R$ that approximate, in a certain sense, the sequence of Frobenius p…
Several authors investigating the asymptotic behaviour of the Betti diagrams of the graded system obtained by taking powers of an ideal have shown that the shape of the nonzero entries in the diagrams stabilizes when $I$ is a homogeneous ideal with generators of the same degree. In this paper, we study the Betti diagrams of graded systems of ideals built by taking the initial ideals or generic initial ideals of powers, and discuss the stabilization of additional collections of Betti diagrams. Our main result shows that when I has generators of the same degree, the entries in the Betti diagrams…
The purpose of this paper is to give affirmative answers to two open questions as follows. Let $(R, \m)$ be a generalized Cohen-Macaulay Noetherian local ring. Both questions, the first question was raised by M. Rogers \cite {R} and the second one is due to S. Goto and H. Sakurai \cite {GS1}, ask whether for every parameter ideal $\q$ contained in a high enough power of the maximal ideal $\m $ the following statements are true: (1) The index of reducibility $N_R(\q;R)$ is independent of the choice of $\q$; and (2) $I^2=\q I$, where $I=\q:_R\m$.
Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $γ> 0$ such that for all $n \ge 0$, $$\ell_R(R/I_{n+1}) - \ell_R(R/I_n) < γn^{d-1}.$$
Asymptotic Behavior of Differential Powers
本文研究了理想上的微分幂操作，探讨了单调理想在特征0下的渐近主理想类，分析了普通幂与微分幂之间的包含关系，并定义了类似积分闭包的微分闭包操作。
Asymptotic behaviour and stability index of v-numbers of graded ideals
本文将Ficarra关于单项式理想幂的v-数猜想推广到分次理想，证明其对主理想、图的覆盖理想、加权有向图边理想等多类理想成立，并引入稳定性指数研究边理想情形。
We define cone structures and asymptotic invariants for multigraded systems of ideals, and show that essentially the only restrictions on such structures is convexity, which is imposed formally.
Let $\mathbb{K}$ be a field and $R = \mathbb{K}[x_1, \ldots, x_n]$. We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in $R$. We study the effect on the resurgence when sum, product and intersection of ideals are taken. We obtain sharp upper and lower bounds for the resurgence and asymptotic resurgence of cover ideals of finite simple graphs in terms of associated combinatorial invariants. We also explicitly compute the resurgence and asymptotic resurgence of cover ideals of several classes of graphs. We characterize a graph being bipartite in terms of t…
Rational symbolic powers of ideals
本文研究了理想有理符号幂的性质，包括其成员标准、二项式展开公式以及单项理想有理符号幂的凸几何描述，并证明了其渐进行为的稳定性。
We specify a class of graphs, $H_t$, and characterize the irreducible decomposition of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question "For each integer $t\geq 0$, does there exist a (hyper) graph $H_t$ such that stabilization of associated primes occurs at $s\geq (χ(H_t)-1)+t$?" asked by Francisco, Hà, and Van Tuyl. For each $t$, $H_t$ has chromatic number 3 and associated primes that stabilize at $s=2+t$.
Asymptotic Properties of Filtrations of Ideals
本文提出了一种通过理想滤波研究交换代数中持久现象的统一框架，并证明了强持久性滤波的性质。
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.
We find explicit bounds on the primary components and on the Castelnuovo-Mumford regularity of powers of monomial ideals. We also analyze the primary decompositions of Katzman's example.
Symbolic Powers and Asymptotic Invariants of GL-Invariant Ideals
该研究针对一般矩阵的线性基变换群作用下的GL不变理想，确定其普通幂、饱和理想、符号幂，给出两类渐近不变量的显式公式，证明相关广义符号Rees代数是Noether的。
Recently, chains of increasing symmetric ideals have attracted considerable attention. In this note, we summarize some results and open problems concerning the asymptotic behavior of several algebraic and homological invariants along such chains, including codimension, projective dimension, Castelnuovo-Mumford regularity, and Betti tables.
Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomi…
Let $\mathscr{I}$ be an ideal sheaf on $P^n$. In the first part of this paper, we bound the asymptotic regularity of powers of $\mathscr{I}$ as $ps-3\leq \reg \mathscr{I}^p\leq ps+e$, where $e$ is a constant and $s$ is the $s$-invariant of $\mathscr{I}$. We also give the same upper bound for the asymptotic regularity of symbolic powers of $\mathscr{I}$ under some conditions. In the second part, by using multiplier ideal sheaves, we give a vanishing theorem of powers of $\mathscr{I}$ when it defines a local complete intersection subvariety with log canonical singularities.
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 …
We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping number is necessarily quasi-monomial. This conjecture holds in dimension two. In general, we reduce it to the case of affine space and to graded sequences of valuation ideals. Along the way, we study the structure of a suitable valuation space.
Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all its powers are integrally closed). For a monomial ideal the skew Waldschmidt constants have an interpretation…
One of the fundamental invariants connecting algebra and geometry is the degree of an ideal. In this paper we derive the probabilistic behavior of degree with respect to the versatile Erdős-Rényi-type model for random monomial ideals defined in \cite{rmi}. We study the staircase structure associated to a monomial ideal, and show that in the random case the shape of the staircase diagram is approximately hyperbolic, and this behavior is robust across several random models. Since the discrete volume under this staircase is related to the summatory higher-order divisor function studied in number …
Recently, Okounkov, Lazarsfeld and Mustata, and Kaveh and Khovanskii have shown that the growth of a graded linear series on a projective variety over an algebraically closed field is asymptotic to a polynomial. We give a complete description of the possible asymptotic growth of graded linear series on projective schemes over a perfect field. If the scheme is reduced, then the growth is polynomial like, but the growth can be very complex on nonreduced schemes. We also give an example of a graded family of m-primary ideals {I_n} in a nonreduced d-dimensional local ring R, such that the length o…
Generic initial ideals (gins in short) were systematically introduced by Galligo in 1974 under the name of Grauert invariants since they appeared apparently first in works of Grauert and Hironaka. Ever since they are of interest in commutative algebra and indirectly in algebraic geometry. Recently Mayes in a series of articles associated to gins geometric objects called limiting shapes. The construction resembles that of Okunkov bodies but there are some differences as well. This work is motivated by Mayes articles and explores the connections between gins, limiting shapes and some asymptotic …
Asymptotic depth of invariant chains of edge ideals
研究不变于递增函数作用的边理想链的渐近深度，揭示相关图的组合与拓扑性质，确定独立复形的渐近同调群行为。
Gotzmann's Persistence states that the growth of an arbitrary ideal can be controlled by comparing it to the growth of the lexicographic ideal. This is used, for instance, in finding equations which cut out the Hilbert scheme (of subschemes of $\mathbf{P}^n$ with fixed Hilbert polynomial) sitting inside an appropriate Grassmannian. We introduce the notion of an {\it extremal ideal} which extends the notion of the lex ideal to other term orders. We then state and prove a version of Gotzmann's theorem for these ideals, valid in an open subset of a Grassmannian.
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 …
Let X = Proj R be a projective scheme over a field k, and let I be an ideal in R generated by forms of the same degree d. Let Y --> X be the blowing up of X along the subscheme defined by I, and let f: Y --> Z be the projection of Y given by the divisor dH - E, where E is the exceptional divisor of the blowup and H is the pullback of a general hyperplane in X. We investigate how the asymptotic linearity of the regularity and a*-invariant of I^q (for q large) is related to invariants of fibers of f.
We find simple necessary and sufficient conditions on a local ring $R$ of dimension $d$ for the limit $$ \lim_{i\rightarrow\infty}\frac{\ell_R(R/I_n)}{n^d} $$ to exist whenever $\{I_n\}$ is a graded family of $m_R$-primary ideals, and give a number of applications. We also give simple necessary and sufficient conditions on projective schemes over a field $k$ for asymptotic limits of the growth of all graded linear series of a fixed Kodaira-Iitaka dimension to exist.
Let $\mathscr{I}$ be an ideal sheaf on $P^n$ defining a subscheme $X$. Associated to $\mathscr{I}$ there are two elementary invariants: the invariant $s$ which measures the positivity of $\mathscr{I}$, and the minimal number $d$ such that $\mathscr{I}(d)$ is generated by its global sections. It is now clear that the asymptotic behavior of $\reg \mathscr{I}^t$ is governed by $s$ but usually not linear. In this paper, we first describe the linear behavior of the asymptotic regularity by showing that if $s=d$, i.e., $s$ reaches its maximal value, then for $t$ large enough $\reg \mathscr{I}^t=dt+e…
We revisit a bound on symbolic powers found by Ein-Lazarsfeld-Smith and subsequently improved by Takagi-Yoshida. We show that the original argument of Ein-Lazarsfeld-Smith actually gives the same improvement. On the other hand, we show by examples that any further improvement based on the same technique appears unlikely. This is primarily an exposition; only some examples and remarks might be new.
We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld \cite{einErmanLazarsfeld16} we give an explicit range in which the graded Betti numbers of $\mathbb{P}^{n_1}\times \mathbb{P}^{n_2}$ embedded by $\mathcal{O}_{\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}}(d_1,d_2)$ are non-zero. These bounds provide the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.
Gotzmann proved the persistence for minimal growth for ideals. His theorem is called Gotzmann's persistence theorem. In this paper, based on the combinatorics on binomial coefficients, a simple combinatorial proof of Gotzmann's persistence theorem in the special case of monomial ideals is given.
Polymatroidal ideals and their asymptotic syzygies
本文研究多拟阵理想幂的同调移位理想的渐近行为，证明了第一同调移位代数的生成性质与强持久性质，提出并部分验证了高阶同调移位代数生成次数及相伴素数递增链的猜想，同时证明了分量多拟阵理想在第一同调移位下的封闭性。
We show that two ostensibly different versions of the asymptotic resurgence introduced by Guardo, Harbourne and Van Tuyl in 2013 are the same. We also show that the resurgence and asymptotic resurgence attain their maximal values simultaneously, if at all, which we apply to a conjecture of Grifo. For radical ideals of points, we show that the resurgence and asymptotic resurgence attain their minimal values simultaneously. In addition, we introduce an integral closure version of the resurgence and relate it to the other versions of the resurgence. In closing we provide various examples and rais…
Optimal Bounds on the Growth of Iterated Sumsets in Abelian Semigroups
本文为阿贝尔半群有限子集的迭代和集增长给出最优上界，证明源自Macaulay定理的界可达，构造基于多项式代数中的单项式理想及其通过Gröbner基的形变。
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