Summary: The Cox ring of an algebraic variety naturally generalizes the homogeneous coordinate ring of the projective space. Projective varieties with finitely generated Cox ring are called Mori dream spaces. In this setting, the Cox ring is a powerful tool for the explicit study of the underlying variety.
After introducing Cox rings and presenting basic algebraic and geometric aspects, the school continues with the following topics: relations to toric geometry, finite generation, varieties with higher complexity torus actions, the surface case and computational aspects of Cox rings and Mori dream spaces.
The prerequisites are basic algebraic geometry, also some knowledge on toric geometry will be helpful. A basic reference is the book Cox Rings by Arzhantsev, Derenthal, Hausen and Laface (Cambridge University Press).

Program of the school: Two lectures each morning, 90 min each, followed by two 60 min excercise sessions in the afternoon.

• Monday September 7:
  • Lecture 1 (JH): Introductory examples of Cox rings, foundations on graded algebras and quasitorus actions, toric varieties.
  • Lecture 2 (AL): Foundations in surface theory, smooth (toric) del Pezzo surfaces
• Tuesday September 8:
  • Lecture 3 (JH): Cox sheaves, Cox rings, characteristic spaces, quotients of H-factorial quasiaffine varieties
  • Lecture 4 (AL): Resolutions, Picard graded Betti numbers, Cox rings of smooth del Pezzo surfaces
• Wednesday September 9:
  • Lecture 5 (JH): Variation of GIT quotients, combinatorial theory of Mori Dream Spaces, Hu/Keel-characterization
  • Lecture 6 (SK): Computational aspects of Mori dream spaces, working with MDSpackage
• Thursday September 10:
  • Lecture 7 (AL): Classification of smooth Mori Dream Surfaces
  • Lecture 8 (SK): Cox rings and blowing up, algorithmic computation of Cox rings
• Friday September 11:
  • Lecture 9 (JH): Cox ring of a variety with torus action, T-varieties of complexity one
  • Lecture 10 (AL): K3 surfaces and k*-surfaces

Organizers: Joachim Jelisiejew, Lukasz Sienkiewicz, Jaroslaw Wisniewski, Institute of Mathematics, the University of Warsaw.

The school is supported by: Warsaw Center of Mathematics and Computer Science, Institute of Mathematics of the University of Warsaw and by research projects Algebraic geometry: varieties and structure (2013/08/A/ST1/00804) and Algebraic varieties: arithmetic and geometry (2012/07/B/ST1/03343) grant of Polish Center of Scientific Research.