Frobenius vectors, Hilbert series and gluings

TL;DR

This paper investigates how key invariants like the Frobenius vector and Hilbert series behave under the gluing of affine semigroups, providing theoretical insights and applications to complete intersection cases.

Contribution

It proves that the Frobenius vector and Hilbert series are preserved under gluings of affine semigroups, extending understanding of their structural properties.

Findings

Frobenius vector is preserved under gluings.

Hilbert series remains invariant when semigroups are glued.

Applications to complete intersection affine semigroups are demonstrated.

Abstract

Let $S_1$ and $S_2$ be two affine semigroups and let $S$ be the gluing of $S_1$ and $S_2$. Several invariants of $S$ are then related to those of $S_1$ and $S_2$; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given.