Secant dimensions of low-dimensional homogeneous varieties

TL;DR

This paper provides a complete description of higher secant dimensions for all connected homogeneous projective varieties of dimension up to 3, utilizing tropical geometry techniques to derive new and more concise results.

Contribution

It offers new calculations for secant dimensions of specific low-dimensional homogeneous varieties, including some with previously unknown results, using a tropical approach.

Findings

Secant dimensions for P^2 * P^1 and F are newly computed.

Results for P^1 * P^1, P^1 * P^1 * P^1, and P^2 * P^1 are more concise than previous proofs.

The tropical method proves effective for calculating secant dimensions.

Abstract

We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all Segre-Veronese embeddings of P^1 * P^1, P^1 * P^1 * P^1, and P^2 * P^1, as well as for the variety F of incident point-line pairs in P^2. For P^2 * P^1 and F the results are new, while the proofs for the other two varieties are more compact than existing proofs. Our main tool is the second author's tropical approach to secant dimensions.