Very general monomial valuations of P2 and a Nagata type conjecture

Marcin Dumnicki; Brian Harbourne; Alex Küronya; Joaquim Roé; Tomasz Szemberg

doi:https://doi.org/10.4310/CAG.2017.v25.n1.a4

Very general monomial valuations of P2 and a Nagata type conjecture

Marcin Dumnicki, Brian Harbourne, Alex Küronya, Joaquim Roé, Tomasz Szemberg

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

6 Citations (Scopus)

Abstract

It is well known that multi-point Seshadri constants for a small number t of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for t ≥ 9 points. Tackling the problem in the language of valuations one can make sense of t points for any real t ≥ 1. We show somewhat surprisingly that a Nagata-type conjecture should be valid for t ≥ 8 + 1/36 points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for t ≤ 7 + 1/9. In the range 7 + 1/9 ≤ t ≤ 8 + 1/36 we are able to compute some sporadic values.

Original language	English
Pages (from-to)	125-161
Journal	Communications in Analysis and Geometry
Volume	25
DOIs	https://doi.org/10.4310/CAG.2017.v25.n1.a4
Publication status	Published - 1 Jan 2017

Keywords

Anticanonical divisor
Monomial valuations
Nagata Conjecture
SHGH Conjecture
Seshadri constants

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abstract = "It is well known that multi-point Seshadri constants for a small number t of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for t ≥ 9 points. Tackling the problem in the language of valuations one can make sense of t points for any real t ≥ 1. We show somewhat surprisingly that a Nagata-type conjecture should be valid for t ≥ 8 + 1/36 points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for t ≤ 7 + 1/9. In the range 7 + 1/9 ≤ t ≤ 8 + 1/36 we are able to compute some sporadic values.",

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T1 - Very general monomial valuations of P2 and a Nagata type conjecture

AU - Dumnicki, Marcin

AU - Harbourne, Brian

AU - Küronya, Alex

AU - Roé, Joaquim

AU - Szemberg, Tomasz

PY - 2017/1/1

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N2 - It is well known that multi-point Seshadri constants for a small number t of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for t ≥ 9 points. Tackling the problem in the language of valuations one can make sense of t points for any real t ≥ 1. We show somewhat surprisingly that a Nagata-type conjecture should be valid for t ≥ 8 + 1/36 points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for t ≤ 7 + 1/9. In the range 7 + 1/9 ≤ t ≤ 8 + 1/36 we are able to compute some sporadic values.

AB - It is well known that multi-point Seshadri constants for a small number t of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for t ≥ 9 points. Tackling the problem in the language of valuations one can make sense of t points for any real t ≥ 1. We show somewhat surprisingly that a Nagata-type conjecture should be valid for t ≥ 8 + 1/36 points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for t ≤ 7 + 1/9. In the range 7 + 1/9 ≤ t ≤ 8 + 1/36 we are able to compute some sporadic values.

KW - Anticanonical divisor

KW - Monomial valuations

KW - Nagata Conjecture

KW - SHGH Conjecture

KW - Seshadri constants

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DO - https://doi.org/10.4310/CAG.2017.v25.n1.a4

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