Weighted Veronese Rings via Convex Semigroups

TL;DR

This paper investigates properties of weighted Veronese rings and convex semigroups, providing algebraic characterizations, examples, and insights into their structure and properties such as Koszulness and Hilbert series.

Contribution

It offers a detailed analysis of two-dimensional normal affine semigroup rings and weighted Veronese rings, including their algebraic and combinatorial properties, with new examples in higher dimensions.

Findings

Determined determinantal presentations and Gr"obner bases for these rings.

Analyzed the graded Hilbert series and Betti numbers.

Provided examples showing certain properties may fail in higher dimensions.

Abstract

We determine properties of two-dimensional normal affine semigroup rings, and in particular of weighted Veronese rings, including determinantal presentation, Gr\"obner basis, graded Hilbert series and graded Betti numbers, the structure of their associated graded rings, and their Koszul property. We give examples in higher dimensions illustrating that the first and last properties may fail. Our approach leverages convex monomial ideals as introduced in Herzog-Qureshi-Saem(2019), which give rise to convex semigroups.