Frobenius numbers and the first Hilbert coefficients of certain numerical semigroup rings

TL;DR

This paper studies the three-generated numerical semigroup $H=\langle a,a+1,a+d\rangle$ and its semigroup ring $R=k[[H]]$. It provides explicit closed forms for the Frobenius number $\mathrm{F}(H)$ in terms of $a$, $d$, and the remainder $r$ from $a=qd+r$, together with formulas for the first Hilbert coefficient $e_1(R)$ derived from genus differences between $H$ and its blow-up $H'$. The results rely on Apéry-set computations and a Herzog-type presentation of the defining ideal, yielding precise conditions under which the formulas hold (e.g., $\gcd(a,d)=1$, $q+r\ge d-2$, and non-symmetry). These findings give computable invariants for a natural class of three-generated semigroups and illuminate the algebraic structure of their defining ideals and related graded rings.

Abstract

Let $a,b$ be positive integers. In this note, we study the numerical semigroup $H=\left<a,a+1,b\right>$ and and the associated numerical semigroup ring $R=k[[H]]$. Under the certain conditions, we provide explicit formulas for the Frobenius number of $H$ and for the first Hilbert coefficient of $R$.

Theorems & Definitions (39)

• Proposition 1.2: see NNW12
• Theorem 1.3: Propositions \ref{['prop37']}+\ref{['prop38']}+\ref{['e1first']}
• Theorem 1.4: Theorem \ref{['mainthm']}+Proposition \ref{['Fro-Chern']}
• Lemma 2.1: RGS09, Lemma 2.4
• Lemma 2.2
• proof
• Proposition 2.3: see Rosal24, Proposition 3.6
• Remark 2.5
• Lemma 2.6
• Lemma 2.7