Betti numbers and linear covers of points

Hailong Dao; Ben Lund; Sreehari Suresh-Babu

Betti numbers and linear covers of points

Commutative Algebra 2024-11-12 v2 Algebraic Geometry Combinatorics

Authors: Hailong Dao , Ben Lund , Sreehari Suresh-Babu

Abstract

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.

Keywords

random matrix ensembles incidence geometry of points and lines

Cite

@article{arxiv.2408.14064,
  title  = {Betti numbers and linear covers of points},
  author = {Hailong Dao and Ben Lund and Sreehari Suresh-Babu},
  journal= {arXiv preprint arXiv:2408.14064},
  year   = {2024}
}

Comments

There is a gap in the key statement 3.2 on matroids that we did not yet know how to fix

Betti numbers and linear covers of points

Abstract

Keywords

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