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

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 $hyp(\mathrm{\Gamma })$ be the maximum number of points of $\mathrm{\Gamma }$ contained in any hyperplane, and let $d(\mathrm{\Gamma })=\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }-hyp(\mathrm{\Gamma })$. If $I\subset R=\mathbb{K}[x_0,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ is the ideal of $\mathrm{\Gamma }$, then in \cite{t1} it is shown that for $n=2,3$, $d(\mathrm{\Gamma })$ has a lower bound expressed in terms of some shift in the graded minimal free resolution of $R\mathrm{/}I$. In these notes we show that this behavior is true in general, for any $n\ge 2$: $d(\Gamma)\geq A_n$,where$A_n=\min{a_i-n}$and$\oplus_i R(-a_i)$ is the last module in the graded minimal free resolution of $R\mathrm{/}I$. In the end we also prove that this bound is sharp for a whole class of examples due to Juan Migliore (\cite{m}).