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On blowing up minimal toric surfaces

Antonio Laface, Luca Ugaglia

Abstract

We prove that the Cox ring of the blowing-up of a minimal toric surface of Picard rank two is finitely generated. As part of our proof of this result we provide a necessary and sufficient condition for finite generation of Cox rings of normal projective $\mathbb Q$-factorial surfaces.

On blowing up minimal toric surfaces

Abstract

We prove that the Cox ring of the blowing-up of a minimal toric surface of Picard rank two is finitely generated. As part of our proof of this result we provide a necessary and sufficient condition for finite generation of Cox rings of normal projective -factorial surfaces.
Paper Structure (11 sections, 19 theorems, 42 equations, 1 figure)

This paper contains 11 sections, 19 theorems, 42 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Toric surfaces
  3. Abelian monoids
  4. Toric Varieties
  5. Toric surfaces
  6. Lattice ideals
  7. Toric surfaces
  8. Minimal toric surfaces
  9. Effective cones
  10. Cox rings
  11. A Magma library

Key Result

Theorem 1

Let $\mathbb P$ be a minimal toric surface of Picard rank two. Then the Cox ring of ${\rm Bl}_e\mathbb P$ is finitely generated.

Figures (1)

  • Figure 1: ${\rm Eff}({\rm Bl}_e\mathbb P)$

Theorems & Definitions (57)

  • Theorem 1
  • Proposition 1.1
  • proof
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Corollary 1.5
  • proof
  • Definition 2.1
  • ...and 47 more