On the identifiability of binary Segre products

TL;DR

This paper proves the conditions under which a product of multiple projective lines embedded via the Segre map is identifiable, meaning each general point on its secant variety uniquely determines a secant space, for sufficiently large m.

Contribution

It establishes new bounds for the identifiability of Segre products of projective lines when embedded in projective space, extending previous understanding.

Findings

Identifiability holds for products of more than 5 copies of P^1.

The paper provides explicit bounds on k for identifiability.

Results apply to secant varieties up to a certain dimension.

Abstract

We prove that a product of $m>5$ copies of $\PP^1$, embedded in the projective space $\PP^r$ by the standard Segre embedding, is $k$-identifiable (i.e. a general point of the secant variety $S^k(X)$ is contained in only one $(k+1)$-secant $k$-space), for all $k$ such that $k+1\leq 2^{m-1}/m$.