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

Abstract

Let K be a field of characteristic 0. Let Γ ⊂ P K n 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 ⊂ R = K [ x 0 , … , x n ] is the ideal of Γ , then in Tohaˇneanu (2009) [12] 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 holds true in general, for any n ≥ 2 : d ( Γ ) ≥ A n , where A n = min { a i − n } and ⊕ 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 (2010) [10].