Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials

A. DICKENSTEIN, M.I HERRERO Y L. TABERA

Barranquilla

Congreso; V CLAM, Sesión de Geometría algebraica; 2016

UMALCA

Severi  varieties are classical objects in algebraicgeometry which give parameter spaces for nodal hypersurfaces. Mikhalkin´scorrespondence theorem from 2005  allowsto compute tropically the degree of the Severi varieties of nodal curves with afixed number of nodes defined by polynomials with support in a given latticepolygon.  The tropical curves appearingin Mikhalkin´s correspondence theorem can be described by the associatedregular subdivision of the support. That is, the set of tropical  curves with a specified combinatorial typecounted in Mikhalkin´s formula, correspond to polyhedral cones in theassociated secondary fan associated with the lattice points in thepolygon.  However, these cones are afraction of all possible cones in the associated tropical Severi variety. E.Katz noted in 2009 that there are maximal cones that are not supported incones  of the secondary fan. Thus, thecombinatorial description of the curves is not enough in many cases to decideif a tropical curve given by a tropical polynomial lies in the correspondingSeveri variety.  This behavior was alsoobserved by J. J. Yang, who gave a partial description of the tropicalizationof the Severi varieties in 2013 and 2016. We explore this phenomenon and give a full characterization in the univariatesetting, that is, we describe all the cones in the tropical Severi varietydefined  by the tropicalization of thevariety 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 valuationsof the elementary symmetric functions of the roots of a polynomial with twodouble roots. Despite its apparent simplicity, the computation of the tropical Severi variety has both combinatorialand arithmetic ingredients. Joint work with Maria Isabel Herrero and  Luis Felipe Tabera.