Betti numbers and linear covers of points

TL;DR

This paper characterizes when the Betti number eta_{n,n+1} of a finite point set in projective space is non-zero, linking it to the geometric configuration of the points lying on the union of two planes.

Contribution

It provides a direct, concise proof connecting Betti numbers of point sets to their geometric arrangement, using combinatorial matroid techniques.

Findings

eta_{n,n+1} is non-zero iff points lie on two planes with combined dimension less than n

The proof is short, direct, and relies on combinatorial matroid properties

Establishes a geometric criterion for Betti number non-vanishing

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.