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# 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: http://dx.doi.org/10.4310/CAG.2017.v25.n1.a4

#### Authors

#### 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

Received 18 April 2015

Published 9 June 2017