Demailly's Conjecture and the containment problem

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

doi:10.1016/j.jpaa.2021.106863

Cited by 8 publications

(7 citation statements)

References 53 publications

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“…In this manuscript, we use Cremoma transformation to get appropriate lower bounds for α(I) of the defining ideals of generic points, and apply the bounds to show the aforementioned inequality, hence, stronger containment for ideals defining small numbers of generic points. After that, a similar approach using specialization as in [BGHN20], see also [BGHN22], yields the results for a small number of general points. Note that the Waldschmidt constant for defining ideals of up to N + 3 generic points are computed in [DHSTG14] and Harbourne-Huneke Containment as well as Chudnovsky's Conjecture would follow easily, see also [NT19].…”

Section: Introductionmentioning

confidence: 92%

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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Section: Introductionmentioning

confidence: 92%

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

Bisui¹,

Nguyên²

2021

Preprint

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“…Two recent preprints, [5,6], focus on the Containment problem and related conjectures. In the first one, the authors show that Chudnovsky's Conjecture holds for sufficiently many general points and to prove it they show that one of the containments conjectured by Harbourne and Huneke holds eventually, meaning for large powers (see Theorem 4.6 in [5]).…”

Section: B Harbourne Conjectured In [4]mentioning

confidence: 99%

“…We also point out that the two above preprints [5,6] do not compute the Waldschmidt constant exactly for any class of ideals, they study lower bounds for the Waldschmidt constant. And, since in [3] the authors found the exact value of the Waldschmidt constant for the Complement of a Steiner configurations of points, then Chudnovsky and Demailly's Conjectures easily follow for our class of ideals (see Section 3).…”

Section: B Harbourne Conjectured In [4]mentioning

confidence: 99%

Steiner Configurations ideals: containment and colouring

Ballico¹,

Favacchio²,

Guardo³

et al. 2021

Preprint

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Given a homogeneous ideal I ⊆ k[x 0 , . . . , xn], the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pair m, r ∈ N, I (m) ⊆ I r holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne -Huneke Conjecture and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in P n k . We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly's Conjectures. Moreover, given a hypergraph H, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to H. We apply these results in the case that H is a Steiner System.

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“…From the intrinsic point of view choosing one of the inflexion points as the origin (in the group law on E) the set of all 9 flex points is exactly the set of 3-division points E [3] on E, i.e., points P subject to the condition 3P = 0 in the addition law on E. This link between the intrinsic and extrinsic geometry is justified by Abel's Theorem [17,Theorem IV.4.13B].…”

Section: Introductionmentioning

confidence: 99%

“…Since flex points of a smooth curve are well known to be common zeroes of the equation f defining this curve and the Hessian H(f ) = H 1 (f ), this provides an extrinsic, concrete way to obtain the ideal of all 3-division points E [3] on E as a complete intersection ideal generated by f and H(f ).…”

Section: Introductionmentioning

confidence: 99%

Sextactic points on the Fermat cubic curve and arrangements of conics

Szemberg¹,

Szpond²

2022

Preprint

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The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of 6-division points on the Fermat cubic F and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction for searching for conceptual proofs of facts stated here and their generalisations. Extensions in several directions seem possible (taking curves of higher degree and contact to F , studying higher degree curves passing through higher order division points on F , studying curves passing through intersection points of already constructed curves, taking the duals etc.) and we hope some younger colleagues might find pleasure in following proposed paths as well as finding their own.

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Demailly's Conjecture and the containment problem

Cited by 8 publications

References 53 publications

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

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

Steiner Configurations ideals: containment and colouring

Sextactic points on the Fermat cubic curve and arrangements of conics

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