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To study syzygies of the Cox rings of del Pezzo surfaces, we calculateimportant syzygetic invariants such as the Hilbert functions, theGreen-Lazarsfeld indices, the projective dimensions, and theCastelnuovo-Mumford regularities. Using these computations as well as thenatural multigrading structures by the Picard groups of del Pezzo surfaces andWeyl group actions on Picard lattices, we determine the Betti diagrams of theCox rings of del Pezzo surfaces of degree at most four.

Hilbert functions of Cox rings of del Pezzo surfaces