贝蒂数和点的线性盖

arXiv - MATH - Commutative Algebra Pub Date : 2024-08-26 DOI:arxiv-2408.14064

Hailong Dao, Ben Lund, Sreehari Suresh-Babu

{"title":"贝蒂数和点的线性盖","authors":"Hailong Dao, Ben Lund, Sreehari Suresh-Babu","doi":"arxiv-2408.14064","DOIUrl":null,"url":null,"abstract":"We prove that for a finite set of points $X$ in the projective $n$-space over\nany field, the Betti number $\\beta_{n,n+1}$ of the coordinate ring of $X$ is\nnon-zero if and only if $X$ lies on the union of two planes whose sum of\ndimension is less than $n$. Our proof is direct and short, and the inductive\nstep rests on a combinatorial statement that works over matroids.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"57 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Betti numbers and linear covers of points\",\"authors\":\"Hailong Dao, Ben Lund, Sreehari Suresh-Babu\",\"doi\":\"arxiv-2408.14064\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove that for a finite set of points $X$ in the projective $n$-space over\\nany field, the Betti number $\\\\beta_{n,n+1}$ of the coordinate ring of $X$ is\\nnon-zero if and only if $X$ lies on the union of two planes whose sum of\\ndimension is less than $n$. Our proof is direct and short, and the inductive\\nstep rests on a combinatorial statement that works over matroids.\",\"PeriodicalId\":501475,\"journal\":{\"name\":\"arXiv - MATH - Commutative Algebra\",\"volume\":\"57 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Commutative Algebra\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.14064\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Commutative Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.14064","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

我们证明，对于任意域的投影 $n$ 空间中的有限点集合 $X$，当且仅当 $X$ 位于其维度之和小于 $n$ 的两个平面的联合面上时，$X$ 的坐标环的贝蒂数 $\beta_{n,n+1}$ 为非零。我们的证明直接而简短，归纳步骤基于对矩阵有效的组合声明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Betti numbers and linear covers of points

We prove that for a finite set of points $X$ in the projective $n$-space over any field, the Betti number $\beta_{n,n+1}$ of the coordinate ring of $X$ is non-zero if and only if $X$ lies on the union of two planes whose sum of dimension is less than $n$. Our proof is direct and short, and the inductive step rests on a combinatorial statement that works over matroids.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

arXiv - MATH - Commutative Algebra

自引率

0.00%

发文量