Secant Varieties of (P ^1) X .... X (P ^1) (n-times) are NOT Defective for n \geq 5

TL;DR

This paper proves that for the Segre embedding of (P^1)^n, the higher secant varieties generally have the expected dimension, with a specific exception for _3(V_4).

Contribution

It establishes that secant varieties of the Segre product of projective lines are non-defective for n  5, identifying a unique defect in _3(V_4).

Findings

_3(V_4) is defective by one dimension.

All other secant varieties _s(V_n) for n  5 are non-defective.

The expected dimension formula holds except for the identified case.

Abstract

Let V_n be the Segre embedding of (P^1) x ... X (P^1) (n times). We prove that the higher secant varieties, \sigma_s(V_n), always have the expected dimension, except for \sigma_3(V_4), which is of dimension 1 less than expected.