On the degree of curves with prescribed multiplicities and bounded negativity

TL;DR

This paper establishes bounds on the degree of plane curves with specified multiplicities at valuation centers and explores implications for the bounded negativity conjecture on rational surfaces.

Contribution

It provides new bounds on curve degrees and Seshadri-type constants related to divisorial valuations, advancing understanding of Nagata-type conjectures and bounded negativity.

Findings

Lower bounds on degrees of curves with prescribed multiplicities

Upper bounds for Seshadri-type constants of valuations

Results supporting the bounded negativity conjecture

Abstract

We provide a lower bound on the degree of curves of the projective plane $\mathbb{P}^2$ passing through the centers of a divisorial valuation $\nu$ of $\mathbb{P}^2$ with prescribed multiplicities, and an upper bound for the Seshadri-type constant of $\nu$, $\hat{\mu}(\nu)$, constant that is crucial in the Nagata-type valuative conjecture. We also give some results related to the bounded negativity conjecture concerning those rational surfaces having the projective plane as a relatively minimal model.