A Tropical Approach to Secant Dimensions
نویسندگان
چکیده
Tropical geometry yields good lower bounds, in terms of certain combinatorialpolyhedral optimisation problems, on the dimensions of secant varieties. 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; this proof might be generalisable to cover all Veronese embeddings, whose secant dimensions are known from the ground-breaking but difficult work of Alexander and Hirschowitz. Also, the non-defectiveness of certain Segre embeddings is proved, which cannot be proved with the rook covering argument already known in the literature. Short self-contained introductions to secant varieties and the required tropical geometry are included.
منابع مشابه
Secant Dimensions of Low-dimensional Homogeneous Varieties
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 P1 × P1, P1 × P1 × P1, and P2 × P1, as well as for the variety F of incident point-line pairs in P2. For P2 × P1 and F the results are new, while the...
متن کاملSecant Dimensions of Minimal Orbits: Computations and Conjectures
We present an algorithm for computing the dimensions of higher secant varieties of minimal orbits. Experiments with this algorithm lead to many conjectures on secant dimensions, especially of Grassmannians and Segre products. For these two classes of minimal orbits, we also point out a relation between the existence of certain codes and non-defectiveness of certain higher secant varieties.
متن کاملOn Degenerate Secant
We propose a class of projective manifolds with degenerate secant varieties, which class is wider than that of Scorza varieties , and study some properties of this class of manifolds. For example, we show that there is a strong restriction on dimensions of manifolds in this class. We also give classiications of such manifolds of low dimensions.
متن کاملInvestigation of Utilizing a Secant Stiffness Matrix for 2D Nonlinear Shape Optimization and Sensitivity Analysis
In this article the general non-symmetric parametric form of the incremental secant stiffness matrix for nonlinear analysis of solids have been investigated to present a semi analytical sensitivity analysis approach for geometric nonlinear shape optimization. To approach this aim the analytical formulas of secant stiffness matrix are presented. The models were validated and used to perform inve...
متن کاملThree notions of tropical rank for symmetric matrices
We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension ...
متن کامل