  To study syzygies of the Cox rings of del Pezzo surfaces, we calculate
important syzygetic invariants such as the Hilbert functions, the
Green-Lazarsfeld indices, the projective dimensions, and the
Castelnuovo-Mumford regularities. Using these computations as well as the
natural multigrading structures by the Picard groups of del Pezzo surfaces and
Weyl group actions on Picard lattices, we determine the Betti diagrams of the
Cox rings of del Pezzo surfaces of degree at most four.
