Chudnovsky's Conjecture and the stable Harbourne-Huneke containment

Bisui, Sankhaneel; Grifo, Eloísa; Harbourne, Brian; Nguyên, Thái Thành

doi:10.48550/arxiv.2004.11213

Cited by 4 publications

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“…Our bound in Theorem 3.3 is sharp. For example, if m = x 2 x 3 x 5 x 6 x 8 x 10 , then we have T (m) = (6, 5, 4, 2) and IT (m) = (10,8,6,3). For this monomial, ℓ = 2 since t 2 = 4 and…”

Section: Andmentioning

confidence: 99%

“…Specifically, the Waldschmidt constant can be used to gain insight into the ideal containment problem, that is, comparing the regular powers of an ideal to those of its symbolic powers. The papers [2,3,4,7] form a small sample of recent work on the Waldschmidt constant; an introduction to this topic can also be found in [6].…”

Section: Introductionmentioning

confidence: 99%

“…In our example, (2 − 1, 3 − 2, 5 − 3, 6 − 4, 8 − 5, 10 − 6) = (1, 1, 2, 2, 3, 4), so t 0 = 6, and t 1 = 5, t 2 = 4 and t 3 = 2 since these are the indices where i j − j < i j+1 − (j + 1). Continuing with this example, the tuple (10,8,6,3) records the indices of the variables where the "jump" occurs.…”

Section: Introductionmentioning

confidence: 99%

“…Example 2.6. In Remark 2.3 it was shown that the monomial m = x 2 x 3 x 5 x 6 x 8 x 10 has T (m) = (6, 5, 4, 2) and IT (m) = (10,8,6,3). So the associated primes of sfBorel(m) are in one-to-one correspondence with the minimal generators of sfBorel({x 2 x 3 , x 4 x 5 x 6 , x 5 x 6 x 7 x 8 , x 6 x 7 x 8 x 9 x 10 }).…”

Section: Introductionmentioning

confidence: 99%

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On the Waldschmidt constant of square-free principal Borel ideals

Camps-Moreno¹,

Kohne²,

Sarmiento³

et al. 2021

Preprint

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Fix a square-free monomial m ∈ S = K[x 1 , . . . , x n ]. The square-free principal Borel ideal generated by m, denoted sfBorel(m), is the ideal generated by all the square-free monomials that can be obtained via Borel moves from the monomial m. We give upper and lower bounds for the Waldschmidt constant of sfBorel(m) in terms of the support of m, and in some cases, exact values. For any rational a b ≥ 1, we show that there exists a square-free principal Borel ideal with Waldschmidt constant equal to a b .

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“…Our bound in Theorem 3.3 is sharp. For example, if m = x 2 x 3 x 5 x 6 x 8 x 10 , then we have T (m) = (6, 5, 4, 2) and IT (m) = (10,8,6,3). For this monomial, ℓ = 2 since t 2 = 4 and…”

Section: Andmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Waldschmidt constant of square-free principal Borel ideals

Camps-Moreno¹,

Kohne²,

Sarmiento³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…, x N . Chudnovsky's conjecture follows from stable version of the containment, which has been studied in [BGHN20].…”

Section: Introductionmentioning

confidence: 99%

Chudnovsky's Conjecture and the stable Harbourne-Huneke containment for general points

Bisui¹,

Nguyên²

2021

Preprint

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We apply Cremona transformation to provide appropriate lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective space. Applying these results, we establish stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of a set of a small number of general points in P 4 . We also prove stable Harbourne-Huneke containment, and thus, Chudnovsky's conjecture for the defining ideal of s general points in P N in the following cases: (a) : N + 4 s N +2 N where N 5; and (b) : 2 N s 3 N where N = 5, 6, 7, and 8.

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Demailly's Conjecture and the Containment Problem

Bisui¹,

Grifo²,

Harbourne³

et al. 2020

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We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.

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Chudnovsky's Conjecture and the stable Harbourne-Huneke containment

Cited by 4 publications

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On the Waldschmidt constant of square-free principal Borel ideals

On the Waldschmidt constant of square-free principal Borel ideals

Chudnovsky's Conjecture and the stable Harbourne-Huneke containment for general points

Demailly's Conjecture and the Containment Problem

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