Secondary fans and tropical Severi varieties

Yang, Jihyeon Jessie

doi:10.1007/s00229-015-0773-3

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2017

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“…Thus, the combinatorial description of the curves is not enough in many cases to decide if a tropical curve given by a tropical polynomial is in the corresponding Severi variety or not. This behavior was also observed by J. J. Yang, who gave a partial description of the tropicalization of the Severi varieties in [17,18]. We explore this phenomenon and give a full characterization in the univariate setting for the case of δ = 2 nodes.…”

Section: Introductionmentioning

confidence: 69%

Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials

2017

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Abstract. We give a description of the tropical Severi variety of univariate polynomials of degree n having two double roots. We show that, as a set, it is given as the union of three explicit types of cones of maximal dimension n − 1, where only cones of two of these types are cones of the secondary fan of {0, . . . , n}. Through Kapranov's theorem, this goal is achieved by a careful study of the possible valuations of the elementary symmetric functions of the roots of a polynomial with two double roots. Despite its apparent simplicity, the computation of the tropical Severi variety has both combinatorial and arithmetic ingredients.

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Section: Introductionmentioning

confidence: 69%