Bounds for syzygies of monomial curves

Abstract

Let Γ ⊆ N \Gamma \subseteq \mathbb {N} be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of Γ \Gamma which depends only on the width of Γ \Gamma , that is, the difference between the largest and the smallest generator of Γ \Gamma . In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width.