A tropical approach to secant dimensions

Tropical geometry yields good lower bounds, in terms of certain combinatorial–polyhedral optimisation problems, on the dimensions of secant varieties. The approach is especially successful for toric varieties such as Segre–Veronese embeddings. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; and indeed, no Segre–Veronese embeddings are known where the tropical lower bound does not give the correct dimension. Short self-contained introductions to secant varieties and the required tropical geometry are included.