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

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

2012-03-09

arxiv.org/abs/1203.2040v1

J. Pure Appl. Algebra 215(2011), 2645-2651

Mathematics

Commutative Algebra

11 pages

Scientific paper

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}).

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