| Résume | Severi varieties are classical objects in algebraic geometry which give parameter spaces for nodal hypersurfaces. Mikhalkin's correspondence theorem from 2005
allows to compute tropically the degree of the Severi varieties of nodal curves with a fixed number of nodes defined by polynomials with support in a given lattice polygon.
The tropical curves appearing in Mikhalkin's correspondence theorem can be described by the associated regular subdivision of the support. That is, the set of tropical
curves with a specified combinatorial type counted in Mikhalkin's formula, correspond to polyhedral cones in the associated secondary fan associated with the lattice points in the polygon.
However, these cones are a fraction of all possible cones in the associated tropical Severi variety. E. Katz noted in 2009 that there are maximal cones that are not supported in cones
of the secondary fan. Thus, the combinatorial description of the curves is not enough in many cases to decide if a tropical curve given by a tropical polynomial lies in the corresponding Severi variety.
This behavior was also observed by J. J. Yang, who gave a partial description of the tropicalization of the Severi varieties in 2013 and 2016.
We explore this phenomenon and give a full characterization in the univariate setting, that is, we describe all the cones in the tropical Severi variety defined
by the tropicalization of the variety of univariate polynomials with fixed degree and two double roots. 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. Joint work with Maria Isabel Herrero and Luis Felipe Tabera. |
