A tropical approach to secant dimensions

By Jan Draisma. 2006

86 citations recorded by the source corpus.

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.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f7ee1f9f20f68e730bd1cdaec81df9a753258da8

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Citations in the corpus

A Tropical Toolkit (2006, methodology): “pical lifting lemma. This lemma was ﬁrst announced without proof in [37]. A proposed proof was given in [34] but has been found to be incomplete. A proof using aﬃnoid algebras was given by Draisma in [9]. Jensen, Markwig, and Markwig provided an algorithm that ﬁnds a tropical lift in [19]. This algo”
A guide to tropicalizations (2011, methodology): “lled the fundamental theorem of tropical algebraic geometry. It is due to Kapranov in the hypersurface case (unpublished manuscript, later incorporated in [EKL]) and to Speyer{Sturmfels [SS], Draisma [Dra], Payne [Pay] in general. Theorem 5.6 For a closed subscheme Xof T, the tropical variety Trop v”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “, [88, 91] for the computational complexity approach, [16, 90] for the biological statistical approach, [92, 22, 94, 93] for the classical algebraic geometry approach, [95, 96] for the representation theory approach, [97] for a tropical approach and [98] for a multilinear algebra approach.”
On the Alexander–Hirschowitz theorem (2007, background): “A proof of Theorem 1.1 in the case n = 2 along these lines was published by Draisma [Dr].”
Induction for secant varieties of Segre varieties (2006, unknown): “G1a1X + G 2 a2X + . . . + G k akX ⊆ T(1,...,1) has dimension D = min{s(1 +”
Equations for Chow and Hilbert quotients (2007, background): “1] and [Einsiedler et al. 2006; Draisma 2008; Jensen et al. 2008; Payne 2009].”
A Field of Generalised Puiseux Series for Tropical Geometry (2007, background): “[Ray74, Thm. 2] and [Rib92, (5.2)] both reduce this fact to general results in the ramification theory of non-archimedian valued fields.”
Geometry of the Restricted Boltzmann Machine (2009, background): “The first tropical RBM model TM(1) n is the image of the tropical secant map for the Segre variety (P(1)) in the sense of Develin [8] and Draisma [10].”