Hilbert functions of Cox rings of del Pezzo surfaces

TL;DR

This paper computes key algebraic invariants like Hilbert functions and Betti diagrams of Cox rings for del Pezzo surfaces, revealing their syzygetic structures and graded properties.

Contribution

It provides explicit calculations of syzygetic invariants and Betti diagrams for Cox rings of del Pezzo surfaces of degree ≤ 4, advancing understanding of their algebraic structure.

Findings

Betti diagrams of Cox rings for degree ≤ 4 del Pezzo surfaces are explicitly determined.

Hilbert functions and Castelnuovo-Mumford regularities are computed for these Cox rings.

The study reveals the influence of Picard group gradings and Weyl group actions on the algebraic invariants.

Abstract

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.