Minimal plane valuations

TL;DR

This paper explores the properties of minimal plane valuations, linking them to conjectures in algebraic geometry, and provides new families of such valuations with potential implications for longstanding conjectures.

Contribution

It proves that the Greuel-Lossen-Shustin Conjecture implies a variation of the Nagata Conjecture involving minimal valuations, extending previous results to all divisorial and irrational valuations.

Findings

Proves the implication of the Greuel-Lossen-Shustin Conjecture for a Nagata-type conjecture involving minimal valuations.

Constructs infinitely many families of very general minimal valuations with multiple Puiseux exponents.

Provides asymptotic evidence supporting the conjecture by Dumnicki et al.

Abstract

We consider the last value $\hat{\mu} (\nu)$ of the vanishing sequence of $H^0(L)$ along a divisorial or irrational valuation $\nu$ centered at $\mathcal{O}_{\mathbb{P}^2,p}$, where $L$ resp. $p$ is a line resp. a point of the projective plane $\mathbb{P}^2$ over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that $\hat{\mu} (\nu) \geq \sqrt{1 / \mathrm{vol}(\nu)}$ and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variation of the Nagata Conjecture involving minimal valuations (that extends the one stated in the paper "Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture" by Dumnicki et al. to the whole set of divisorial and irrational valuations of the projective plane) which also implies the original one. We also provide infinitely many families of very general minimal valuations with an arbitrary number of Puiseux exponents, and an asymptotic result that can be considered as an evidence in the direction of the mentioned conjecture by Dumnicki et al.