Semicontinuity and X-rank

Abstract

Let X ⊂ ℙ^r be an integral and non-degenerate variety. For each q ∈ ℙ^r, the X-rank r_X(q) of q is the minimal cardinality of a subset of X whose linear span contains q. Obviously, r_X^-1(1) = X. In this note, we give conditions which assure that r_X^-1(2) ∪ X is not closed in ℙ^r and hence the function r_X is not semicontinuous.