Minimal decomposition of binary forms with respect to tangential projections

TL;DR

This paper characterizes the minimal decomposition (rank) of points on a cuspidal curve obtained via tangential projection of a rational normal curve, linking it to schemes computing ranks on the original curve.

Contribution

It provides a new description of the rank of points on the projected curve in terms of schemes related to the original rational normal curve.

Findings

Explicit formula for the rank of points on the projected curve.

Connection between the projected curve's rank and schemes computing the original curve's rank.

Insight into minimal decompositions with respect to tangential projections.

Abstract

Let $C\subset \mathbb{P}^n$ be a rational normal curve and let $\ell_O:\mathbb{P}^{n+1}\dashrightarrow \mathbb{P}^n$ be any tangential projection form a point $O\in T_AC$ 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 $B$.