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Cox rings of nef anticanonical rational surfaces

Michela Artebani, Sofía Pérez Garbayo

Abstract

This paper deals with the problem of computing a generating set for the Cox ring $R(X)$ of a smooth projective rational surface $X$ with nef anticanonical class. In case $R(X)$ is finitely generated, we show that the degrees of its generators are either classes of negative curves, elements of the Hilbert basis of the nef cone or certain ample classes of anticanonical degree one, which only appear when $X$ is a rational elliptic surface of Halphen index $m>2$. Moreover, we partially characterize which elements of the Hilbert basis of the nef cone are irredundant for generating $R(X)$. We apply this result to compute explicit minimal generating sets for Cox rings of some rational elliptic surfaces of Halphen index $>1$.

Cox rings of nef anticanonical rational surfaces

Abstract

This paper deals with the problem of computing a generating set for the Cox ring of a smooth projective rational surface with nef anticanonical class. In case is finitely generated, we show that the degrees of its generators are either classes of negative curves, elements of the Hilbert basis of the nef cone or certain ample classes of anticanonical degree one, which only appear when is a rational elliptic surface of Halphen index . Moreover, we partially characterize which elements of the Hilbert basis of the nef cone are irredundant for generating . We apply this result to compute explicit minimal generating sets for Cox rings of some rational elliptic surfaces of Halphen index .
Paper Structure (11 sections, 18 theorems, 53 equations, 1 table)

This paper contains 11 sections, 18 theorems, 53 equations, 1 table.

Table of Contents

  1. Preliminaries
  2. Anticanonical surfaces
  3. Cox rings
  4. Main Theorem
  5. Examples
  6. Halphen rational elliptic surfaces of type E8
  7. A 2-Halphen rational elliptic surface of type D8
  8. A 2-Halphen rational elliptic surface of type E7+A1
  9. A 2-Halphen rational elliptic surface of type A8
  10. A weak del Pezzo surface of Picard number 7
  11. Weak del Pezzo surfaces dominated by extremal jacobian elliptic surfaces

Key Result

Theorem 1

Let $X$ be a smooth projective rational surface such that $-K_X$ is nef and whose Cox ring $R(X)$ is finitely generated. The degrees of a minimal homogeneous generating set of $R(X)$ belong to the following list:

Theorems & Definitions (39)

  • Theorem
  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 1.6: B*Lemma I.5
  • Lemma 1.7: ACL
  • ...and 29 more