Extreme values of the resurgence for homogeneous ideals in polynomial rings

Harbourne, Brian; Kettinger, Jake; Zimmitti, Frank

doi:10.1016/j.jpaa.2021.106811

Cited by 7 publications

(5 citation statements)

References 19 publications

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“…We have learned that Harbourne, Kettinger, Zimmitti in [40] and DiPasquale, Drabkin in [19] proved independently that ρ a (I) < h if and only if ρ(I) < h. As pointed out in [19] Remark 2.3, the next result is similar as Proposition 4.1.3 of Denkert's thesis [18], as Lemma 4.12 in [20] and as Proposition 2.2 in [19].…”

Section: Asymptotic Resurgence and Stable Harbourne Conjecturesupporting

confidence: 60%

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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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Section: Asymptotic Resurgence and Stable Harbourne Conjecturesupporting

confidence: 60%

“…It is clear from the definition that 1 ≤ ρ a (I) ≤ ρ(I). As pointed out in [40], DiPasquale, Francisco, Mermin, Schweig showed that ρ a (I) = sup{m/r : I (m) I r }, where I r is the integral closure of I r (see also [20] Corollary 4.14) .…”

Section: B Harbourne Conjectured In [4]mentioning

confidence: 92%

Steiner Configurations ideals: containment and colouring

Ballico¹,

Favacchio²,

Guardo³

et al. 2021

Preprint

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“…Remark 5.20. Corollary 5.19 can also be recovered by the recent Theorem 2.3 in [19] where the authors provide various examples and questions about computing the resurgence of homogeneous ideals.…”

mentioning

confidence: 88%

“…We also compute the Waldschmidt constant and the resurgence of the ideal defining a Hadamard fat grid (see Proposition 5.17 and Corollary 5. 19).…”

Section: Hadamard Fat Gridsmentioning

confidence: 99%

Hadamard products of symbolic powers and Hadamard fat grids

Jafarloo¹,

Bocci²,

Guardo³

et al. 2022

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In this paper we address the question if, for points P, Q ∈ P 2 , I(P ) m I(Q) n = I(P Q) m+n−1 and we obtain different results according to the number of zero coordinates in P and Q. Successively, we use our results to define the so called Hadamard fat grids, which are the result of the Hadamard product of two sets of collinear points with given multiplicities. The most important invariants of Hadamard fat grids, as minimal resolution, Waldschmidt constant and resurgence, are then computed.

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“…follows from [15] and [18] that the stable Harbourne conjecture is true when the resurgence or the asymptotic resurgence is strictly smaller than the bigheight h. A great source of examples for which the resurgence and asymptotic resurgence are known, comes either from the geometric side ( [4,5,10,24,25]) or from combinatorial side (see [30,28,19,29]).…”

Section: Introductionmentioning

confidence: 99%

On the resurgence and asymptotic resurgence of homogeneous ideals

Jayanthan¹,

Kumar²,

Mukundan³

2021

Preprint

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Let K be a field and R = K[x 1 , . . . , x n ]. We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in R. We study the effect on the resurgence when sum, product and intersection of ideals are taken. We obtain sharp upper and lower bounds for the resurgence and asymptotic resurgence of cover ideals of finite simple graphs in terms of associated combinatorial invariants. We also explicitly compute the resurgence and asymptotic resurgence of cover ideals of several classes of graphs. We characterize a graph being bipartite in terms of the resurgence and asymptotic resurgence of edge and cover ideals. We also compute explicitly the resurgence and asymptotic resurgence of edge ideals of some classes of graphs.

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Extreme values of the resurgence for homogeneous ideals in polynomial rings

Cited by 7 publications

References 19 publications

Steiner Configurations ideals: containment and colouring

Steiner Configurations ideals: containment and colouring

Hadamard products of symbolic powers and Hadamard fat grids

On the resurgence and asymptotic resurgence of homogeneous ideals

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