Buchsbaumness of finite complement simplicial affine semigroups

TL;DR

This paper classifies Buchsbaum simplicial affine semigroups with finite complements, establishing a characterization based on gaps and pseudo-Frobenius elements, and explores their structural properties and invariants.

Contribution

It provides a complete classification of such semigroups, characterizes Buchsbaum property via gaps and pseudo-Frobenius elements, and analyzes their structural and numerical properties.

Findings

Buchsbaum semigroups are characterized by their gaps and pseudo-Frobenius elements.

Explicit formulas are given for the minimal presentation in certain cases.

Buchsbaum property is not preserved under gluing, unlike other properties.

Abstract

In this article, we classify all Buchsbaum simplicial affine semigroups whose complement in their (integer) rational polyhedral cone is finite. We show that such a semigroup is Buchsbaum if and only if its set of gaps is equal to its set of pseudo-Frobenius elements. Furthermore, we provide a complete structure of these affine semigroups. In the case of affine semigroups with maximal embedding dimension, we provide an explicit formula for the cardinality of the minimal presentation in terms of the number of extremal rays, the embedding dimension, and the genus. Finally, we observe that, unlike the complete intersection, Cohen-Macaulay, and Gorenstein properties, the Buchsbaum property is not preserved under gluing of affine semigroups.