Seshadi constants via functions on Newton–Okounkov bodies

Dumnicki, Marcin; Küronya, Alex; Maclean, Catriona; Szemberg, Tomasz

doi:10.1002/mana.201500280

Cited by 6 publications

(6 citation statements)

References 12 publications

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“…Relating the Seshadri constants and Okounkov bodies is not new, see e.g. [Ito13,DKMS16]. The key insight here is to use the infinitesimal approach in the form which is a slight generalization of [KurLoz14, Example 4.4].…”

Section: An Application Of Okounkov Bodies To the Seshadri Constantsmentioning

confidence: 99%

Restrictions on Seshadri Constants on Surfaces

Farnik¹,

Szemberg²,

Szpond³

et al. 2017

Taiwanese J. Math.

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Starting with the pioneering work of Ein and Lazarsfeld [EinLaz93] restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors [Bau99, BauSze11, HarRoe08, KSS09, Nak05, Ste98, Sze12, Xu95].In the present note we show how approximation involving continued fractions combined with recent results of Küronya and Lozovanu on Okounkov bodies of line bundles on surfaces [KurLoz14, KurLoz15] lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong additional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number 1.

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Section: An Application Of Okounkov Bodies To the Seshadri Constantsmentioning

confidence: 99%

Restrictions on Seshadri Constants on Surfaces

Farnik¹,

Szemberg²,

Szpond³

et al. 2017

Taiwanese J. Math.

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“…This is [33,Theorem 4.14 and Proposition 5.6] In lucky cases invariants of the functions ϕ F will not depend on the domain ∆ v (L), or more precisely, the choice of Y • or v. In this case they give rise to asymptotic invariants of the line bundle L. Proof. The first number is independent of the choice of the flag by [17], Proposition 2.2 and Theorem 2.4 (note that both proofs go through verbatim in the current setting), the second one is [5,…”

Section: Bmentioning

confidence: 99%

Concave transforms of filtrations and rationality of Seshadri constants

Küronya¹,

Maclean²,

Roé³

2019

Preprint

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We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton-Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded algebra is the algebra of sections of a concrete line bundle in higher dimension. We use this description to give a rationality criterion for certain Seshadri constants. Along the way we introduce Newton-Okounkov bodies of abstract graded semigroups and determine conditions for their slices to be Newton-Okounkov bodies of subsemigroups. 75980-P, while the first author also enjoyed partial support from the NKFI Grant No. 115288 'Algebra and Algorithms', and the third also from AGAUR 2017SGR585. Our project was initiated during the workshop 'Newton-Okounkov Bodies, Test Configurations, and Diophantine Geometry' at the Banff International Research Station. We appreciate the stimulating atmosphere and the excellent working conditions at BIRS.

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“…These bodies will be defined later and are convex sets of R 2 . Our development is supported on [85,Chapter 5], [8], [86], [82] [15], [40] and [38]. Here, we use the notation established before, although in this section k = C.…”

Section: Seshadri-type Constants and Newton-okounkov Bodiesmentioning

confidence: 99%

“…Another important invariant related to the Seshadri constants is given in [40]. Let D be an big divisor on Z 0 .…”

Section: Seshadri-type Constants Of Divisorial Valuationsmentioning

confidence: 99%