Cox rings of projective varieties

Abstract

The Cox ring of a smooth algebraic projective variety X is the direct sum of H^0(X,D) as D varies in the Picard group of X. The ring was introduced by Y. Hu and S. Keel in 2000 and generalizes Cox' construction of the homogenous coordinate ring of a toric variety. The main reason for studying these rings is that Cox(X) is finitely generated if and only if X is a so-called Mori dream space, and this has strong implications for the birational geometry of X. The aim of this thesis is to survey some properties of Cox rings and present some new results about Cox rings of rational surfaces and threefolds occurring as blow-ups of P^2 and P^3 and some special K3 surfaces of Picard number two.