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On the degree of curves with prescribed multiplicities and bounded negativity

Carlos Galindo, Francisco Monserrat, Carlos-Jesús Moreno-Ávila, Elvira Pérez-Callejo

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 $ν$ of $\mathbb{P}^2$ with prescribed multiplicities, and an upper bound for the Seshadri-type constant of $ν$, $\hatμ(ν)$, 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.

On the degree of curves with prescribed multiplicities and bounded negativity

Abstract

We provide a lower bound on the degree of curves of the projective plane passing through the centers of a divisorial valuation of with prescribed multiplicities, and an upper bound for the Seshadri-type constant of , , 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.

Paper Structure

This paper contains 7 sections, 10 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Divisorial valuations
  3. Plane divisorial valuations
  4. Divisorial valuations of $\mathbb{P}^2$
  5. Non-positive at infinity special divisorial valuations of $\mathbb{F}_\delta$
  6. A lower bound on the degree of a curve of $\mathbb{P}^2$ with prescribed multiplicities
  7. Asymptotic approach to bounded negativity

Key Result

Theorem \oldthetheorem

Let $\nu$ be a special divisorial valuation of $\mathbb{F}_\delta$ centered at a point $p$. Set $Y_\nu$ the rational surface that $\nu$ defines. Then the following conditions are equivalent:

Theorems & Definitions (24)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 14 more