On the X-rank with respect to linear projections of projective varieties

In this paper we improve the known bound for the X-rank R X (P ) of an element P ∈ PN in the case where X ⊂ Pn is a projective variety obtained as a linear projection from a general v-dimensional subspace V ⊂ Pn + v . Then, if X ⊂ Pn is a curve obtained from a projection of a rational normal curve C ⊂ Pn + 1 from a point O ⊂ Pn + 1 , we are able to describe the precise value of the X-rank for those points P ∈ Pn such that R X (P ) ≤ R C (O) − 1 and to improve the general result. We also give a stratification, via the X-rank, of the osculating spaces to projective cuspidal projective curves X. We give a description and a new bound of the X-rank of subspaces both in the general case and with respect to integral non-degenerate projective curves. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim