Asymptotic linearity of regularity and $a^*$-invariant of powers of ideals

Harbourne, Brian

doi:10.4310/mrl.2011.v18.n1.a1

Cited by 18 publications

(15 citation statements)

References 19 publications

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“…the minimal number such that I is integral over I ≤d , where I ≤d denotes the ideal generated by the forms in I of degree at most d. The other coefficient, e, is more mysterious. In the case where I is m-primary and generated by general forms of a single degree, I determines a finite morphism φ : Proj(S) → P r , and [EH10] showed that e+ 1 is the maximum regularity of a fibre of φ; following a conjecture of [Hà10], [Char10] extended this result to the case in which I is generated by arbitrary forms of a single degree in terms of the associated map from the blowup of Proj(S) along the subscheme defined by I. This is quite a tantalizing phenomenon, but so far no similar interpretation has presented itself in a more general case.…”

Section: Introductionmentioning

confidence: 99%

Regularity defect stabilization of powers of an ideal

Berlekamp

2012

Mathematical Research Letters

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Abstract. When I is an ideal of a standard graded algebra S with homogeneous maximal ideal m, it is known by the work of several authors that the Castelnuovo-Mumford regularity of I m ultimately becomes a linear function dm + e for m 0. We give several constraints on the behavior of what may be termed the regularity defect (the sequence e m = reg I m − dm) in various cases. When I is m-primary we give a family of bounds on the first differences of the e m , including an upper bound on the increasing part of the sequence; for example, we show that the e i cannot increase for i ≥ dim(S). When I is a monomial ideal, we show that the e i become constant for i ≥ n(n − 1)(d − 1), where n = dim(S).

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Section: Introductionmentioning

confidence: 99%

Regularity defect stabilization of powers of an ideal

Berlekamp

2012

Mathematical Research Letters

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“…Many recent studies are devoted to investigating these constants (cf. [4,10,18,19,25]). Even for edge ideals of graphs the exact values for these constants are still out of reach.…”

Section: Open Problems and Questionsmentioning

confidence: 99%

Regularity of Squarefree Monomial Ideals

Harbourne

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.Dedicated to Tony Geramita, a great teacher, colleague and friend. IntroductionCastelnuovo-Mumford regularity (or simply regularity) is an important invariant in commutative algebra and algebraic geometry that governs the computational complexity of ideals, modules and sheaves. Computing or finding bounds for the regularity is a difficult problem. Many simply stated questions and conjectures have been verified only in very special cases. For instance, the Eisenbud-Goto conjecture, which states that the regularity of a projective variety is bounded by the difference between its degree and codimension, has been proved only for arithmetic Cohen-Macaulay varieties, curves and surfaces.The class of squarefree monomial ideals is a classical object in commutative algebra, with strong connections to topology and combinatorics, which continues to inspire much of current research. During the last few decades, advances in computer technology and speed of computation have drawn many researchers' attention toward problems and questions involving this class of ideals. Investigating the regularity of squarefree monomial ideals has evolved to be a highly active research topic in combinatorial commutative algebra.In this paper, we survey a number of recent studies on the regularity of squarefree monomial ideals. Even for this class of ideals, the list of results on regularity is too large to exhaust. Our focus will be on studies that find bounds and/or compute the regularity of squarefree monomial ideals in terms of combinatorial data from associated simplicial complexes and hypergraphs. Our aim is to provide readers with an adequate overall picture of the problems, results and techniques, and to demonstrate similarities and differences between these studies. At the same time, we hope to motivate interested researchers, and especially graduate students, to start working in this area. We shall showcase results in a timeline to exhibit (if possible) trends in developing new and/or more general results from previous ones.The paper is structured as follows. In the next section we collect basic notation and terminology from commutative algebra and combinatorics that will be used in the paper. This section provides readers who are new to the research area the necessary background to start. More advanced readers can skip this section and go directly to Section 3. In Section 1 arXiv:1310.7912v2 [math.AC]

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“…To find the exact form of the linear function is also not easy (cf. [1,6,11,38]). In the following, we will call them the linearization-of-regularity problems, by abuse of terminology.…”

Section: Introductionmentioning

confidence: 99%

Regularity of powers of (parity) binomial edge ideals

Shen¹,

Zhu²

2021

Preprint

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In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an almost complete intersection ideal I which is generated by a homogeneous d-sequence. As applications, when I is an almost complete intersection, taking the form of the (parity) binomial edge ideal of a connected graph, we can describe explicitly formulas for reg(I t ) for t ≥ 2. The only exception is when I is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.

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Asymptotic linearity of regularity and $a^*$-invariant of powers of ideals

Cited by 18 publications

References 19 publications

Regularity defect stabilization of powers of an ideal

Regularity defect stabilization of powers of an ideal

Regularity of Squarefree Monomial Ideals

Regularity of powers of (parity) binomial edge ideals

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