On the valuative Nagata conjecture

TL;DR

This paper explores the valuative Nagata conjecture by establishing equivalent conditions for minimal plane divisorial valuations using valuative Seshadri constants and other global surface tools, leading to new equivalent statements and related results.

Contribution

It introduces new equivalent conditions for minimal divisorial valuations and derives several equivalent formulations of the valuative Nagata conjecture.

Findings

Equivalent conditions for minimal divisorial valuations established

New formulations of the valuative Nagata conjecture derived

Connections between valuations, Seshadri constants, and global surface properties

Abstract

We provide several equivalent conditions for a plane divisorial valuation of a smooth projective surface to be minimal with respect to an ample divisor. These conditions involve a valuative Seshadri constant and other global tools of the surface defined by the divisorial valuation. As a consequence, we derive several equivalent statements for the valuative Nagata conjecture and some related results.