The symbolic defect of an ideal

Galetto, Federico; Geramita, Anthony V.; Shin, Yong-Su; Tuyl, Adam Van

doi:10.1016/j.jpaa.2018.11.019

Cited by 27 publications

(29 citation statements)

References 36 publications

(51 reference statements)

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“…Arsie and Vatne study the Hilbert function of I (m) /I m in [1], giving examples for ideals of coordinates of planes and set of points in P n . More recently, in [8] Galetto, Geramita, Shin, and Van Tuyl have studied this module by defining the m-th symbolic defect,…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Drabkin

Guerrieri

2020

Journal of Pure and Applied Algebra

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Let I be an ideal whose symbolic Rees algebra is Noetherian. For m ≥ 1, the m-th symbolic defect, sdefect(I, m), of I is defined to be the minimal number of generators of the module 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 is Noetherian. MSC: 13F20; 05C25.

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Section: Introductionmentioning

confidence: 99%

“…In [8], the first tow questions are analyzed for the defining ideal of a general set of points in P 2 or a set of points forming a star configuration. Interesting results about the symbolic defect of edge ideals of graphs can also be found in the recent work [14].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Drabkin

Guerrieri

2020

Journal of Pure and Applied Algebra

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“…A recent paper of Galetto, Geramita, Shin, and Van Tuyl [9] introduced the notion of symbolic defect to measure the difference between the symbolic power I (t) and ordinary power I t . For a given m, the symbolic defect sdefect(I, m) is the number µ(m) of minimal generators F 1 , F 2 , .…”

Section: Applications To Ideal Containment Questionsmentioning

confidence: 99%

“…This is known as the symbolic defect, and the symbolic defect sequence is the sequence {sdefect(I, m)} m∈N . In [9], the authors study the symbolic defect sequences of star configurations in P n k and homogeneous ideals of points in P 2 k . Our work considers all these questions in the context of a class of edge ideals.…”

Section: Introductionmentioning

confidence: 99%

Comparing powers of edge ideals

2019

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Given a nontrivial homogeneous ideal I ⊆ k[x 1 , x 2 , . . . , x d ], a problem of great recent interest has been the comparison of the rth ordinary power of I and the mth symbolic power I (m) . This comparison has been undertaken directly via an exploration of which exponents m and r guarantee the subset containment I (m) ⊆ I r and asymptotically via a computation of the resurgence ρ(I), a number for which any m/r > ρ(I) guarantees I (m) ⊆ I r . Recently, a third quantity, the symbolic defect, was introduced; as I t ⊆ I (t) , the symbolic defect is the minimal number of generators required to add to I t in order to get I (t) .We consider these various means of comparison when I is the edge ideal of certain graphs by describing an ideal J for which I (t) = I t + J. When I is the edge ideal of an odd cycle, our description of the structure of I (t) yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

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“…Various invariants have been defined to compare symbolic and ordinary powers of ideals: the resurgence [BH10], the Waldschmidt constant [BH10], and symbolic defect [GGST16], among others. These can be in some cases explicitly computed and in others approximated using the SymbolicPowers package.…”

Section: Introductionmentioning

confidence: 99%

Calculations involving symbolic powers

Drabkin

Grifo

Seceleanu

et al. 2019

J. Softw. Alg. Geom.

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Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously difficult to compute, and there are seemingly simple questions related to symbolic powers that remain open even over polynomial rings. In this paper, we describe a Macaulay2 software package that allows for computations of symbolic powers of ideals and which can be used to study the equality and containment problems, among others.2010 Mathematics Subject Classification. Primary 13P99. Secondary: 13A15, 13C99.

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The symbolic defect of an ideal

Cited by 27 publications

References 36 publications

Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Comparing powers of edge ideals

Calculations involving symbolic powers

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