Tropicalization of singular varieties

MARÍA ISABEL HERRERO; ALICIA DICKENTEIN; LUIS TABERA

Cabo Frio

Conferencia; II Latin American School of Algebraic Geometry and Applications - ELGA II; 2015

CIMPA

In this lecture I will present one of my topics of research, the tropicalization of singular varieties. I will mention here some results that we have obtained with Alicia Dickenstein and Luis Tabera, as well as the ongoing research with Alicia.Our main result is that the tropicalization of the Severi varieties given by univariate polinomials of degree n and 2 double roots can be written as the union of three types of cones of (pure) dimension n-1, where 2 of these types are cones in the secondary fan of the support set of the polynomials but the third type is not, it comes from a subdivision with adicional arithmetic restrictions. I will present each of these types, and a breef explanation on how to obtain them from subdivisions.Also, I will present the onging work computing combinatorially the multiplicities of the maximal cones in our description based on the ideas of Yang and Esterov, and the desired future direction of our work.