The Cox ring of a variety is the total coordinate ring, i.e., the direct sum of all spaces of global sections of all divisors. When this ring is finitely generated, the variety is called Mori dream (MD). A necessary condition for being MD is the finite generatedness of Pic(X), i.e., the vanishing of the irregularity. Smooth rational surfaces with big anticanonical divisor are MD. So are all del Pezzo surfaces of any degree. A K3 surface or an Enriques surface with Picard number at least 3 is MD iff its automorphism group is finite.
In this talk I will consider the case of surfaces of general type with pg=0, and provide several examples that are MD. I will also provide non-minimal examples that are not MD. This is a joint work with Kyoung-Seog Lee.