230 research outputs found
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 P^1 * P^1, P^1 * P^1 * P^1, and P^2 * P^1, as well
as for the variety F of incident point-line pairs in P^2. For P^2 * P^1 and F
the results are new, while the proofs for the other two varieties are more
compact than existing proofs. Our main tool is the second author's tropical
approach to secant dimensions.Comment: 25 pages, many picture
Tropical secant graphs of monomial curves
The first secant variety of a projective monomial curve is a threefold with
an action by a one-dimensional torus. Its tropicalization is a
three-dimensional fan with a one-dimensional lineality space, so the tropical
threefold is represented by a balanced graph. Our main result is an explicit
construction of that graph. As a consequence, we obtain algorithms to
effectively compute the multidegree and Chow polytope of an arbitrary
projective monomial curve. This generalizes an earlier degree formula due to
Ranestad. The combinatorics underlying our construction is rather delicate, and
it is based on a refinement of the theory of geometric tropicalization due to
Hacking, Keel and Tevelev.Comment: 30 pages, 8 figures. Major revision of the exposition. In particular,
old Sections 4 and 5 are merged into a single section. Also, added Figure 3
and discussed Chow polytopes of rational normal curves in Section
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
Phylogenetic Algebraic Geometry
Phylogenetic algebraic geometry is concerned with certain complex projective
algebraic varieties derived from finite trees. Real positive points on these
varieties represent probabilistic models of evolution. For small trees, we
recover classical geometric objects, such as toric and determinantal varieties
and their secant varieties, but larger trees lead to new and largely unexplored
territory. This paper gives a self-contained introduction to this subject and
offers numerous open problems for algebraic geometers.Comment: 15 pages, 7 figure
Minkowski sums and Hadamard products of algebraic varieties
We study Minkowski sums and Hadamard products of algebraic varieties.
Specifically we explore when these are varieties and examine their properties
in terms of those of the original varieties.Comment: 25 pages, 7 figure
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