Cox Rings of Rational Surfaces

Cox Rings of Rational Surfaces

January-February, 2014 — Concepción (Chile) and IMPA (Brasil)

Abstract
I will begin with an introduction to Cox rings in general: motivation, definitions, basic properties, first examples. I will then specialize to the case of Cox rings of rational surfaces and focus on the question of finite generation. The first main case is the case of del Pezzo surfaces. I will give a set of generators for the Cox ring and I will also explain the geometric meaning of the quadratic relations. I will outline an argument to prove that the quadratic relations generate the ideal of relations. After this more explicit approach, I will talk about the relationship between finite generation and positivity properties of the anti-canonical divisor. I will give complete criteria for finite generation for surfaces with big anti-canonical and rational elliptic surfaces. I will also present some partial results in the case of pseudo-effective surfaces.

Prerequisites
A basic knowledge of algebraic geometry: blow ups and the classification of rational surfaces. Some notions of Picard groups (only for rational surfaces), intersection pairing, effective cone and nef cone. I will review most of these notions, but some previous familiarity will be expected.

Overview

The minicourse will consist of 10 hours, divided into five lectures, each of approximately two hours. One of the hours develops some of the theory and the outline the main results. The other hour is devoted to examples and motivation.
The rough plan of the lectures is as follows.

  1. First lecture: Cox rings.
    Introduction to Cox rings: toric varieties, Fano and log-Fano varieties.
    Examples and a few computations.

  2. Second lecture: Surfaces.
    Surfaces: (finite generation) \( \Longleftrightarrow \) (Eff is polyhedral and (nef \( \Longrightarrow \) semiample)).
    Examples: rational surfaces with infinitely many exceptional curves, Harbourne's examples.
    Effective cone and nef cone of del Pezzo surfaces.

  3. Third lecture: del Pezzo Surfaces.
    del Pezzo surfaces, quadratic relations and presentations; blow ups of \( \mathbb{P}^2 \) in at most 8 points.
    Tor and Betti numbers, combinatorial games on the graph of curves.
    Possible description of generators for blow ups in few points.

  4. Fourth lecture: Big Rational Surfaces.
    Anticanonical Iitaka dimension 2: big rational surfaces.
    Examples and questions.

  5. Fifth lecture: Not so big rational surfaces.
    Anticanonical Iitaka dimension 0 or 1: elliptic and pseudo-effective rational surfaces.
    Examples of Artebani-Laface, and Laface-T.
    Redundant blow-ups (Hwang-Park).

References

  1. Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen and Antonio Laface; Cox Rings. A free preview is available on the arXiv.
    A book on Cox rings in general.
  2. David Cox; The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17–50.
    David Cox's original paper, introducing Cox rings for toric varieties.
  3. Yi Hu, Sean Keel; Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.
  4. Victor V. Batyrev, Oleg N. Popov; The Cox ring of a del Pezzo surface, in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 85–103, Progr. Math., 226, Birkhäuser Boston, Boston, MA, 2004.
  5. Ana-Maria Castravet, Jenia Tevelev; Hilbert's 14th problem and Cox rings, Compos. Math. 142 (2006), no. 6, 1479–1498.
  6. Mike Stillman, Damiano Testa, Mauricio Velasco; Gröbner bases, monomial group actions, and the Cox rings of del Pezzo surfaces, J. Algebra 316 (2007), no. 2, 777–801.
  7. Antonio Laface, Mauricio Velasco; A survey on Cox rings, Geom. Dedicata 139 (2009), 269–287.
  8. Damiano Testa, Anthony Várilly-Alvarado, Mauricio Velasco; Cox rings of degree one del Pezzo surfaces, Algebra Number Theory 3 (2009), no. 7, 729–761.
  9. Michela Artebani, Jürgen Hausen, Antonio Laface; On Cox rings of K3 surfaces, Compos. Math. 146 (2010), no. 4, 964–998.
  10. Jürgen Hausen, Hendrik Süß; The Cox ring of an algebraic variety with torus action, Adv. Math. 225 (2010), no. 2, 977–1012.
  11. Michela Artebani, Antonio Laface; Cox rings of surfaces and the anticanonical Iitaka dimension, Adv. Math. 226 (2011), no. 6, 5252–5267.
  12. Damiano Testa, Anthony Várilly-Alvarado, Mauricio Velasco; Big rational surfaces, Math. Ann. 351 (2011), no. 1, 95–107.
  13. John Christian Ottem; On the Cox ring of \( \mathbb{P}^2 \) blown up in points on a line, Math. Scand. 109 (2011), no. 1, 22–30.
  14. Dongseon Hwang, Jinhyung Park; Redundant blow-ups and Cox rings of rational surfaces, available on the arXiv.