The cone conjecture for Calabi-Yau pairs in dimension two

Abstract

We prove the Morrison-Kawamata cone conjecture for klt Calabi-Yau pairs in dimension 2. That is, for a large class of rational surfaces as well as K3 surfaces and abelian surfaces, the action of the automorphism group of the surface on the convex cone of ample divisors has a rational polyhedral fundamental domain. More concretely: there many be infinitely many curves with negative self-intersection on the surface, but all such curves fall into finitely many orbits under the automorphism group of the surface. The proof uses the geometry of groups acting on hyperbolic space. We deduce a characterization of the surfaces in this class with finitely generated Cox ring.

22 pages; references added

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• On the number of minimal models of a log smooth threefold
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• Around and beyond the canonical class
• The movable cone of Calabi--Yau threefolds in ruled Fano manifolds
• A cone conjecture for log Calabi-Yau surfaces