A Generalization of the Strong Castelnuovo Lemma

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 ai(X) = dimk[Tor R i (A, k)]i+1. Green’s Strong Castelnuovo Lemma (SCL) shows that if the points are in general position, then an−1(X) 6= 0 if and only if the points are on a rational normal curve. Cavaliere, Rossi and Valla conjectured in [2] that if the points are not necessarily in general position the possible extension of the SCL should be the following: an−1(X) 6= 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.