Surfaces with Big Anticanonical Class

Mori dream spaces were introduced by Y. Hu and S. Keel [9]; they are natural generalizations of toric varieties. We recall the definition. Let X be a Q-factorial and normal projective variety, such that Pic(X)⊗Z Q = N(X). Let D1, . . . , Dr be a collection of divisors that give a basis for Pic(X), and whose affine hull contains the pseudoeffective cone. The Cox ring of X is the multi-graded section ring