128 research outputs found
Three notions of tropical rank for symmetric matrices
We introduce and study three different notions of tropical rank for symmetric
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
of each secant set, the convex hull of the variety, and in most cases, the
smallest secant set which is equal to the convex hull.Comment: 23 pages, 3 figure
Fano schemes of determinants and permanents
Let and denote the subschemes of
given by the determinants (respectively the permanents)
of an matrix of indeterminates. In this paper, we study the
geometry of the Fano schemes and
parametrizing the -dimensional planes in
lying on and , respectively. We
prove results characterizing which of these Fano schemes are smooth,
irreducible, and connected; and we give examples showing that they need not be
reduced. We show that always has the expected
dimension, and we describe its components exactly. Finally, we give a detailed
study of the Fano schemes of -planes on the determinantal and
permanental hypersurfaces.Comment: 43 pages; v2 minor revisions. To appear in AN
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