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

Abstract

Let $\mathbb{K}$ be a field of characteristic 0. Let $\Gamma \subset {\mathbb{P}}_{\mathbb{K}}^n$ 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 $n\ge 2$: $d(\Gamma )\ge A_n$, 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}).

11 pages