Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture

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

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arXiv · 1312.5549

Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture

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

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Abstract

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

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Marcin Dumnicki, Brian Harbourne, Alex Küronya, Joaquim Roé, Tomasz Szemberg. 2016-02-05. Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture. https://arxiv.org/abs/1312.5549

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