Betti numbers and linear covers of points
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910691618717696 |
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| author | Dao, Hailong Lund, Ben Suresh-Babu, Sreehari |
| author_facet | Dao, Hailong Lund, Ben Suresh-Babu, Sreehari |
| contents | We prove that for a finite set of points $X$ in the projective $n$-space over any field, the Betti number $β_{n,n+1}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14064 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Betti numbers and linear covers of points Dao, Hailong Lund, Ben Suresh-Babu, Sreehari Commutative Algebra Algebraic Geometry Combinatorics We prove that for a finite set of points $X$ in the projective $n$-space over any field, the Betti number $β_{n,n+1}$ 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. |
| title | Betti numbers and linear covers of points |
| topic | Commutative Algebra Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2408.14064 |