A generalization of the Strong Castelnuovo Lemma

TL;DR

This paper proves a conjecture extending Green's Strong Castelnuovo Lemma, characterizing when a set of points in projective space lies on a rational normal curve or a union of two linear subspaces based on algebraic invariants.

Contribution

The authors prove a conjecture that generalizes the Strong Castelnuovo Lemma to points not necessarily in general position, linking algebraic invariants to geometric configurations.

Findings

Confirmed the conjecture relating algebraic invariants to geometric configurations.

Extended the characterization of point sets on rational normal curves.

Linked algebraic properties to unions of linear subspaces.

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 $a_i(X)=\dim_k [{\rm Tor}_i^R(A,k)]_{i+1}$. Green's Strong Castelnuovo Lemma (SCL) shows that if the points are in general position, then $a_{n-1}(X)\neq 0$ if and only if the points are on a rational normal curve. Cavaliere, Rossi and Valla conjectured that if the points are not necessarily in general position the possible extension of the SCL should be the following: $a_{n-1}(X)\neq 0$ 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.