A stable version of Harbourne's Conjecture and the containment problem for space monomial curves

Grifo, Eloísa

doi:10.1016/j.jpaa.2020.106435

Cited by 18 publications

(23 citation statements)

References 32 publications

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“…We prove in general that their resurgence is strictly smaller than their codimension, and consequently all of them satisfy the stable Harbourne conjecture. Grifo had previously proved that

P^{(3)} \subseteq P^{2}

for the defining ideals for space monomial curves [19, Theorem 4.1]. In Section 5, we generalize this latter fact to ideals defined by the

2 \times 2

minors of a

2 \times 3

matrix in a regular ring such that the ideal of entries of the matrix is generated by at most five elements.…”

Section: Introductionmentioning

confidence: 80%

“…Our method to prove that P (3) ⊆ mP 2 for all such P is a brute force calculation of the degrees of each generator of the third symbolic power (these ideals are quasi-homogeneous) to show that no such generator can also be a minimal generator of P 2 . Since Grifo [19] proved that P (3) ⊆ P 2 , this degree calculation suffices.…”

Section: Space Monomial Curvesmentioning

confidence: 99%

“…In this paper, we build off recent work by the first author [19], which studied what is called the stable Harbourne conjecture. A full discussion of this conjecture is contained in that paper; here we highlight some of the relevant facts.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, as discussed in [19], there are no counterexamples known to the stable version of Harbourne's conjecture, namely whether

I^{(c n - c + 1)} \subseteq I^{n}

must hold for all

n

sufficiently large. In fact, this stable version of Harbourne's Conjecture holds for very general and generic point configurations in

P^{n}

[43, Theorem 2.2, Remark 2.3], and more general sufficient conditions for the stable Harbourne Conjecture are given in [19]. An important invariant in the comparison of symbolic and usual powers was introduced by Bocci and Harbourne: Definition The resurgence of an ideal

I

is given by

ρ false(I false) = sup \{\frac{m}{s} | I^{false(m false)} \neg \subseteq I^{s}\} .

…”

Section: Introductionmentioning

confidence: 99%

“…In Section 5, we generalize this latter fact to ideals defined by the

2 \times 2

minors of a

2 \times 3

matrix in a regular ring such that the ideal of entries of the matrix is generated by at most five elements. This result is based on work of Seceleanu [38, Theorem 3.3] as extended by Grifo [19, Theorem 3.12].…”

Section: Introductionmentioning

confidence: 99%

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Expected resurgences and symbolic powers of ideals

Grifo

Huneke

Mukundan

2020

J. London Math. Soc.

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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. One 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.

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“…We prove in general that their resurgence is strictly smaller than their codimension, and consequently all of them satisfy the stable Harbourne conjecture. Grifo had previously proved that