A tropical approach to secant dimensions

Abstract

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.

Dimension of Marginals of Kronecker Product Models

“…This case has been covered previously in Draisma's tropical approach to secant dimensions [17]. The corresponding implications for the dimension of (not tropical) mixtures of independence models have also been studied in algebraic geometry and tensor analysis; see [10,2,26].…”

“…The generalization to arbitrary visible hierarchical models is left for future work. Furthermore, similarly to [17], the tropical approach leads in many cases to combinatorial conditions that can be very difficult to verify outside of well-established cardinality bounds for error correcting codes. We think that a promising direction is the formulation of dimension bounds in terms of simpler models, as done in Lemma 25 for the case of a binary independence hidden model.…”

“…Proposition 4 can be regarded as a version of the Bieri-Groves theorem [7,17], stating that the dimension of the marginal model is bounded below by the dimension of its tropical version:…”

“…This is the central idea of [17] and subsequently [14,31] for estimating the dimension of secant varieties and RBMs.…”

Faithful tropicalisation and torus actions

“…This can be taken either as a definition of Trop(X ) or as a theorem when other definitions are chosen [9,11,19,20]. Indeed, in [19] it is proved that X an is the projective limit of the tropicalisations Trop(X ) for all choices of coordinates.…”