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

Let $\mathbb K$ be a field of characteristic 0. Let $Γ\subset\mathbb P^n_{\mathbb K}$ be a reduced finite set of points, not all contained in a hyperplane. Let $hyp(Γ)$ be the maximum number of points of $Γ$ contained in any hyperplane, and let $d(Γ)=|Γ|-hyp(Γ)$. If $I\subset R=\mathbb K[x_0,...,x_n]$ is the ideal of $Γ$, then in \cite{t1} it is shown that for $n=2,3$, $d(Γ)$ 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 $n\geq 2$: $d(Γ)\geq A_n$, 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}).

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