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Nef divisors of surfaces given by pencils at infinity

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

Abstract

We give generators for the nef cone and the cone of curves of rational surfaces obtained by blowing-up the complex projective plane at a set of points $\mathcal{B} \cup \mathcal{D}$, where $\mathcal{B}$ is the set of (proper and infinitely near) base points of a pencil associated with a curve having one place at infinity, and $\mathcal{D}$ is a set of finitely many infinitely near free points on the strict transforms of curves of the pencil. We also prove that, when the pencil is given by an AMS-type curve and $\mathcal{D}$ contains at most two free points on any curve considered, the Cox ring of the obtained surface is finitely generated.

Nef divisors of surfaces given by pencils at infinity

Abstract

We give generators for the nef cone and the cone of curves of rational surfaces obtained by blowing-up the complex projective plane at a set of points , where is the set of (proper and infinitely near) base points of a pencil associated with a curve having one place at infinity, and is a set of finitely many infinitely near free points on the strict transforms of curves of the pencil. We also prove that, when the pencil is given by an AMS-type curve and contains at most two free points on any curve considered, the Cox ring of the obtained surface is finitely generated.
Paper Structure (6 sections, 9 theorems, 54 equations, 2 figures)

This paper contains 6 sections, 9 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Curves and pencils at infinity
  4. Surfaces obtained by eliminating base points of pencils at infinity.
  5. The nef cone of a surface given by configurations extending the pencils at infinity
  6. Rational surfaces whose Cox ring is finitely generated

Key Result

Lemma 1

Let $\varphi,\phi$ be two germs of curve on $\mathbb{P}^2$ at $p$ with no irreducible components in common. Then, it holds that where $q$ runs over all infinitely near to $p$ points that lie on the strict transforms $\widetilde{\varphi}$ and $\widetilde{\phi}$ of both $\varphi$ and $\phi$.

Figures (2)

  • Figure 1: Dual graph $\Gamma_\pi$.
  • Figure 2: Dual graph of $\mathcal{C}$

Theorems & Definitions (23)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • Example 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Theorem 7
  • ...and 13 more