Communications in Analysis and Geometry

Volume 25 (2017)

Number 1

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

Pages: 125 – 161

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

Authors

Marcin Dumnicki (Department of Mathematics, Jagiellonian University, Kraków, Poland)

Brian Harbourne (Department of Mathematics, University of Nebraska, Lincoln, Neb., U.S.A.)

Alex Küronya (Institut für Mathematik, Goethe-Universität Frankfurt, Germany)

Joaquim Roé (Departament de Matemàtiques, Facultat de Ciències, Universitat Autonòma de Barcelona, Spain)

Tomasz Szemberg (Department of Mathematics, Pedagogical University of Cracow, Kraków, Poland)

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 $t \geq 9$ points. Tackling the problem in the language of valuations one can make sense of t points for any real $t \geq 1$. We show somewhat surprisingly that a Nagata-type conjecture should be valid for $t \geq 8 + 1/36$ points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for $t \leq 7 + 1/9$. In the range $7 + 1/9 \leq t \leq 8 + 1/36$ we are able to compute some sporadic values.

Keywords

Nagata conjecture, SHGH conjecture, Seshadri constants, monomial valuations, anticanonical divisor

2010 Mathematics Subject Classification

13A18, 14C20

Full Text (PDF format)

Received 18 April 2015

Published 9 June 2017