Tropical secant graphs of monomial curves

Details Tropical secant graphs of monomial curves Tropical secant graphs of monomial curves

2010-05-19

arxiv.org/abs/1005.3364v2

Mathematics

Algebraic Geometry

30 pages, 8 figures. Major revision of the exposition. In particular, old Sections 4 and 5 are merged into a single section. A

Scientific paper

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.

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