A generalization of the Strong Castelnuovo Lemma

Abstract

We consider a set X of distinct points in the n-dimensional projective space over an algebraically closed field k. Let A denote the coordinate ring of X, and let a i ( X ) = dim k [ Tor i R ( A , k ) ] i + 1 . Green's Strong Castelnuovo Lemma (SCL) shows that if the points are in general position, then a n − 1 ( X ) ≠ 0 if and only if the points are on a rational normal curve. Cavaliere, Rossi and Valla (1995) conjectured in [2] that if the points are not necessarily in general position the possible extension of the SCL should be the following: a n − 1 ( X ) ≠ 0 if and only if either the points are on a rational normal curve or in the union of two linear subspaces whose dimensions add up to n. In this work we prove the conjecture.