Quadratic numerical semigroups and the Koszul property

TL;DR

This paper investigates the conditions under which the associated graded rings of numerical semigroups are quadratic and Koszul, providing bounds, classifications, and characterizations for various classes of semigroups.

Contribution

It offers effective bounds for the multiplicity of semigroups with quadratic associated graded rings and classifies Koszul complete intersection semigroups via gluings, extending to several known classes.

Findings

Bounds for multiplicity when gr_m K[H] is quadratic

Classification of Koszul complete intersection semigroups

Identification of Koszul semigroups among various classes

Abstract

Let $H$ be a numerical semigroup. We give effective bounds for the multiplicity $e(H)$ when the associated graded ring $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadrics. We classify Koszul complete intersection semigroups in terms of gluings. Furthermore, for several classes of numerical semigroups considered in the literature (arithmetic, compound, special almost complete intersections, $3$-semigroups, symmetric or pseudo-symmetric $4$-semigroups) we classify those which are Koszul.