Bounds for syzygies of monomial curves

Caviglia, Giulio; Moscariello, Alessio; Sammartano, Alessio

doi:10.1090/proc/16862

Bounds for syzygies of monomial curves
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by Giulio Caviglia, Alessio Moscariello and Alessio Sammartano;

Proc. Amer. Math. Soc. 152 (2024), 3665-3678

DOI: https://doi.org/10.1090/proc/16862

Published electronically: July 26, 2024

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Abstract:

Let $\Gamma \subseteq \mathbb {N}$ be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of $\Gamma$ which depends only on the width of $\Gamma$, that is, the difference between the largest and the smallest generator of $\Gamma$. 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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Proceedings of the American Mathematical Society

Bounds for syzygies of monomial curves
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Abstract: