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

TL;DR

This paper advances understanding of the X-rank for points relative to projected varieties, providing new bounds, precise rank values for certain points, and a stratification of osculating spaces for cuspidal curves.

Contribution

It improves bounds for X-rank in projected varieties, characterizes ranks for points related to rational normal curves, and introduces a stratification of osculating spaces for cuspidal curves.

Findings

Improved bounds for X-rank in projected varieties.

Exact X-rank values for points near rational normal curve projections.

Stratification of osculating spaces for cuspidal curves.

Abstract

In this paper we improve the known bound for the $X$-rank $R_{X}(P)$ of an element $P\in {\mathbb{P}}^N$ in the case in which $X\subset {\mathbb P}^n$ is a projective variety obtained as a linear projection from a general $v$-dimensional subspace $V\subset {\mathbb P}^{n+v}$. Then, if $X\subset {\mathbb P}^n$ is a curve obtained from a projection of a rational normal curve $C\subset {\mathbb P}^{n+1}$ from a point $O\subset {\mathbb P}^{n+1}$, we are able to describe the precise value of the $X$-rank for those points $P\in {\mathbb P}^n$ such that $R_{X}(P)\leq R_{C}(O)-1$ and to improve the general result. Moreover we give a stratification, via the $X$-rank, of the osculating spaces to projective cuspidal projective curves $X$. Finally 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.