Abstract
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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M. A. Cueto was supported by a UC Berkeley Chancellor’s Fellowship and by an NSF postdoctoral fellowship DMS-1103857 and S. Lin was supported by a Singapore A*STAR Fellowship.
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Cueto, M.A., Lin, S. Tropical secant graphs of monomial curves. Beitr Algebra Geom 54, 383–418 (2013). https://doi.org/10.1007/s13366-012-0091-9
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DOI: https://doi.org/10.1007/s13366-012-0091-9