The minimum distance of sets of points and the minimum socle degree

TL;DR

This paper generalizes a lower bound on the minimum distance of point sets in projective space, linking it to the minimal free resolution of their defining ideals, and proves the bound's sharpness for certain cases.

Contribution

It extends previous results to all dimensions n≥2, establishing a universal lower bound on the minimum distance based on the minimal free resolution.

Findings

The lower bound d(Γ) ≥ A_n holds for all n ≥ 2.

The bound is sharp for specific classes of examples.

The result connects algebraic invariants with geometric properties.

Abstract

Let $\mathbb K$ be a field of characteristic 0. Let $\Gamma\subset\mathbb P^n_{\mathbb K}$ be a reduced finite set of points, not all contained in a hyperplane. Let $hyp(\Gamma)$ be the maximum number of points of $\Gamma$ contained in any hyperplane, and let $d(\Gamma)=|\Gamma|-hyp(\Gamma)$. If $I\subset R=\mathbb K[x_0,...,x_n]$ is the ideal of $\Gamma$, then in \cite{t1} it is shown that for $n=2,3$, $d(\Gamma)$ has a lower bound expressed in terms of some shift in the graded minimal free resolution of $R/I$. In these notes we show that this behavior is true in general, for any $n\geq 2$: $d(\Gamma)\geq A_n$, where $A_n=\min\{a_i-n\}$ and $\oplus_i R(-a_i)$ is the last module in the graded minimal free resolution of $R/I$. In the end we also prove that this bound is sharp for a whole class of examples due to Juan Migliore (\cite{m}).