Bounds for syzygies of monomial curves

Pub Date : 2024-04-03 DOI:10.1090/proc/16862

Giulio Caviglia, Alessio Moscariello, Alessio Sammartano

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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.

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单项式曲线对称性的界限

让 Γ ⊆ N \Gamma \subseteq \mathbb {N} 是一个数字半群。在本文中，我们证明了 Γ \Gamma 的半群环的贝蒂数的上界，它只取决于 Γ \Gamma 的宽度，即 Γ \Gamma 的最大生成器和最小生成器之间的差值。这样，我们在实现赫尔佐格和斯塔马特的猜想方面取得了进展[《代数学杂志》418 (2014)，第 8-28 页]。此外，对于 4 代数值半群--第一个重要的开放情形--我们证明了赫尔佐格-斯塔马特对除有限多个宽度值之外的所有宽度值的约束。

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