Abstract
In this paper we improve the known bound for the X‐rank R~X~(P) of an element \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$P\in {\mathbb {P}}^N$\end{document} in the case where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$X\subset {\mathbb P}^n$\end{document} is a projective variety obtained as a linear projection from a general v‐dimensional subspace \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$V\subset {\mathbb P}^{n+v}$\end{document}. Then, if \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$X\subset {\mathbb P}^n$\end{document} is a curve obtained from a projection of a rational normal curve \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$C\subset {\mathbb P}^{n+1}$\end{document} from a point \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$O\subset {\mathbb P}^{n+1}$\end{document}, we are able to describe the precise value of the X‐rank for those points \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$P\in {\mathbb P}^n$\end{document} 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.