  Let X be a smooth projective variety with torsion-free Picard group. We
introduce complexes of vector spaces whose homology determines the structure of
the minimal free resolution of the Cox ring of X over the polynomial ring and
show how the homology of these complexes can be studied by purely geometric
methods. As an application of these techniques we give a simple new proof of a
characterization of the Cox rings of Del Pezzo surfaces (of degree >1)
conjectured by Batyrev and Popov.
