On the X-rank with respect to linearly normal curves

TL;DR

This paper investigates the X-rank of points relative to smooth linearly normal curves in projective space, establishing bounds and specific cases for genus 2 and higher, with implications for understanding tensor decompositions.

Contribution

It provides new bounds on the X-rank of points with respect to linearly normal curves and offers a complete description for genus 2 cases, extending understanding of tensor ranks.

Findings

X-rank of a general point is bounded by n+1-s under certain conditions

Complete characterization of X-rank for genus 2 curves when n=3,4

Analysis of X-rank for points on the tangential variety when n≥5

Abstract

In this paper we study the $X$-rank of points with respect to smooth linearly normal curves $X\subset \PP n$ of genus $g$ and degree $n+g$. We prove that, for such a curve $X$, under certain circumstances, the $X$-rank of a general point of $X$-border rank equal to $s$ is less or equal than $n+1-s$. In the particular case of $g=2$ we give a complete description of the $X$-rank if $n=3,4$; while if $n\geq 5$ we study the $X$-rank of points belonging to the tangential variety of $X$.