Bounds for syzygies of monomial curves

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

Giulio Caviglia, Alessio Moscariello, Alessio Sammartano

{"title":"Bounds for syzygies of monomial curves","authors":"Giulio Caviglia, Alessio Moscariello, Alessio Sammartano","doi":"10.1090/proc/16862","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma subset-of-or-equal-to double-struck upper N\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Gamma \\subseteq \\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which depends only on the width of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that is, the difference between the largest and the smallest generator of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16862","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

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.

查看原文

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文