Minimal decomposition of binary forms with respect to tangential projections

Abstract

Let $C\subset {\mathbb{P}}^n$ be a rational normal curve and let ${\ell }_O:{\mathbb{P}}^{n+1}⇢{\mathbb{P}}^n$ be any tangential projection form a point $O\in T_{A}C$ where $A\in C$. Hence $X:={\ell }_O(C)\subset {\mathbb{P}}^n$ is a linearly normal cuspidal curve with degree $n+1$. For any $P={\ell }_O(B)$, $B\in {\mathbb{P}}^{n+1}$, the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\subset X$ whose linear span contains $P$. Here we describe $r_X(P)$ in terms of the schemes computing the $C$-rank or the border $C$-rank of $B$.

7 pages

References in corpus (3)

• Computing symmetric rank for symmetric tensors
• Decomposition of homogeneous polynomials with low rank
• On the X-rank with respect to linear projections of projective varieties