Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series

TL;DR

This paper investigates how much the Hilbert series of orbifold del Pezzo surfaces reveals about their singularities, providing nonexistence results, bounds on singularities, and applications to lattice polytope combinatorics.

Contribution

It analyzes the limitations of the Hilbert series in determining singularities of orbifold del Pezzo surfaces and derives bounds and nonexistence results.

Findings

Identifies cases where the Hilbert series cannot determine singularities.

Provides bounds on the number of singularities based on the Hilbert series.

Applies results to lattice polytope combinatorics in the toric setting.

Abstract

The Hilbert series of a polarised algebraic variety $(X,D)$ is a powerful invariant that, while it captures some features of the geometry of $(X,D)$ precisely, often cannot recover much information about its singular locus. This work explores the extent to which the Hilbert series of an orbifold del Pezzo surface fails to pin down its singular locus, which provides nonexistence results describing when there are no orbifold del Pezzo surfaces with a given Hilbert series, supplies bounds on the number of singularities on such surfaces, and has applications to the combinatorics of lattice polytopes in the toric case.