An asymptotic bound for secant varieties of Segre varieties

TL;DR

This paper establishes an asymptotic lower bound for the size of non-defective secant varieties of Segre varieties, providing insights into their defectivity and explicit computational results for specific cases.

Contribution

It introduces an asymptotic lower estimate for non-defective secant varieties of Segre varieties based on the number of factors, advancing understanding of their defectivity.

Findings

Lower bound depends only on the number of factors

Non-defectivity of secant varieties for (P^n)^4 with n between 2 and 10

Explicit computational verification of non-defectivity in specific cases

Abstract

This paper studies the defectivity of secant varieties of Segre varieties. We prove that there exists an asymptotic lower estimate for the greater non-defective secant variety (without filling the ambient space) of any given Segre variety. In particular, we prove that the ratio between the greater non-defective secant variety of a Segre variety and its expected rank is lower bounded by a value depending just on the number of factors of the Segre variety. Moreover, in the final section, we present some results obtained by explicit computation, proving the non-defectivity of all the secant varieties of Segre varieties of the shape (P^n)^4, with 1 < n < 11, except at most \sigma_199((P^8)^4) and \sigma_357((P^10)^4).