On the stratification by $X$-ranks of a linearly normal elliptic curve $X\subset \mathbb {P}^n$

TL;DR

This paper provides a detailed description of how points in projective space are stratified according to their $X$-rank with respect to a linearly normal elliptic curve, enhancing understanding of rank stratification.

Contribution

It offers an almost complete characterization of the stratification of projective space by $X$-rank and open $X$-rank for linearly normal elliptic curves.

Findings

Describes the stratification of $P^n$ by $X$-rank.

Provides conditions for points of specific $X$-ranks.

Analyzes the open $X$-rank stratification.

Abstract

Let $X\subset \mathbb {P}^n$ be a linearly normal elliptic curve. For any $P\in \mathbb {P}^n$ the $X$-rank of $P$ is the minimal cardinality of a set $S\subset X$ such that $P\in \langle S\rangle$. In this paper we give an almost complete description of the stratification of $\mathbb {P}^n$ given by the $X$-rank and the open $X$-rank.