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Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

Carlos Galindo, Francisco Monserrat, Carlos-Jesús Moreno-Ávila

Abstract

We consider rational surfaces $Z$ defined by divisorial valuations $ν$ of Hirzebruch surfaces. We introduce the concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when $ν$ is non-positive at infinity, the extremal rays of the cone of curves of $Z$ can be explicitly given.

Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

Abstract

We consider rational surfaces defined by divisorial valuations of Hirzebruch surfaces. We introduce the concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when is non-positive at infinity, the extremal rays of the cone of curves of can be explicitly given.

Paper Structure

This paper contains 7 sections, 13 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hirzebruch surfaces
  4. Divisorial valuations.
  5. Non-positive and negative at infinity valuations.
  6. The sign at infinity of special valuations
  7. The sign at infinity of non-special valuations

Key Result

Proposition \oldthetheorem

Assume that $\delta \geq 1$. Then there exists a $\delta$-dimensional family of irreducible curves of degree $(0,1)$ passing through a general point of $\mathbb{F}_{\delta}$. Also, a curve of degree $(1,0)$ and an irreducible curve of degree $(0,1)$ meet at a general point of $\mathbb{F}_{\delta}$.

Theorems & Definitions (33)

  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 23 more