Harbourne Constants and Arrangements of Lines on Smooth
            Hypersurfaces in $\mathbb{P}^{3}_{\mathbb{C}}$

Pokora, Piotr

doi:10.11650/tjm.20.2016.6338

Cited by 10 publications

(13 citation statements)

References 11 publications

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“…This phenomenon was already observed in [8,Example 3.4]. However, in that example one has that the H-constant decreases like −d/2 as d → +∞, whereas for the degree d-Schur surface we have…”

Section: Moreover This Bound Is Sharp and It Is Achieved By The Consupporting

confidence: 67%

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Local Negativity of Surfaces With Non-Negative Kodaira Dimension and Transversal Configurations of Curves

Laface¹,

Pokora²

2019

Glasgow Math. J.

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We give a bound on the H-constants of configurations of smooth curves having transversal intersection points only on an algebraic surface of non-negative Kodaira dimension. We also study in detail configurations of lines on smooth complete intersections X ⊂ P n+2 C of multi-degree d = (d1, . . . , dn), and we provide a sharp and uniform bound on their H-constants, which only depends on d. P H(X; P),

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Section: Moreover This Bound Is Sharp and It Is Achieved By The Consupporting

confidence: 67%

“…In this section, we will prove that there exists a uniform bound on Harbourne constants for line configurations on smooth hypersurfaces of degree d in P 3 C for any d ≥ 4. It is worth mentioning that the first non-trivial case d = 3 was fully described in [8,Proposition 4.1].…”

Section: H-constants For Line Configurations On Smooth Complete Intermentioning

confidence: 99%

Local Negativity of Surfaces With Non-Negative Kodaira Dimension and Transversal Configurations of Curves

Laface¹,

Pokora²

2019

Glasgow Math. J.

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“…One can show that α(I(V )) = 21, and hence by Example 1.3.6 that α(I(3Z)) ≤ 21, so α(I(Z)) ≤ 7 by Example 1.3.4(e). 45 , we see that the least m > 0 such that dim[I(mZ)] m15 > 0 is m = 3, and that α(I(Z)) ≤ 15 and α(I(3Z)) ≤ 45.…”

Section: Detailsmentioning

confidence: 84%

Asymptotics of linear systems, with connections to line arrangements

Harbourne

2018

Banach Center Publ.

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The main focus of these notes is recent work on linear systems in which line arrangements play a role, including problems such as semi-effectivity, containment problems of symbolic powers of homogeneous ideals in their powers, bounded negativity, and a new perspective on the SHGH Conjecture. Along the way we will be concerned with asymptotic invariants such as Waldschmidt constants, resurgences and H-constants. Acknowledgements: My participation at the Spring, 2016 miniPAGES was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. Some of the revisions to these notes were done at the Banach Center in Warsaw in the weeks after this material was presented at the workshop at the Pedagogical University in Krakow. I also thank J. Szpond for her helpful comments, and the referee for generously working though these notes and providing copious, helpful feedback.

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“…Let X be a smooth hypersurface of degree d ≥ 3 in P 3 . The Harbourne constants for line arrangements on X were first studied in [18]. The bounds obtained there were generalized in [15].…”

Section: Introductionmentioning

confidence: 99%

Bounded negativity and Harbourne constants on ruled surfaces

Hanumanthu

Subramaniam

2020

manuscripta math.

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Let X be a smooth projective surface and let C be an arrangement of curves on X. The Harbourne constant of C was defined as a way to investigate the occurrence of curves of negative self-intersection on blow ups of X. This is related to the bounded negativity conjecture which predicts that the self-intersection number of all reduced curves on a surface is bounded below by a constant. We consider a geometrically ruled surface X over a smooth curve and give lower bounds for the Harbourne constants of transversal arrangements of curves on X. We also define a global Harbourne constant as the infimum of Harbourne constants for arrangements of a specific type and give a lower bound for it.

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Harbourne Constants and Arrangements of Lines on Smooth Hypersurfaces in $\mathbb{P}^{3}_{\mathbb{C}}$

Abstract: In this note we find a bound for the so-called linear Harbourne constants for smooth hypersurfaces in P 3 C .

Cited by 10 publications

References 11 publications

Local Negativity of Surfaces With Non-Negative Kodaira Dimension and Transversal Configurations of Curves

Local Negativity of Surfaces With Non-Negative Kodaira Dimension and Transversal Configurations of Curves

Asymptotics of linear systems, with connections to line arrangements

Bounded negativity and Harbourne constants on ruled surfaces

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