Higher Connectivity of Tropicalizations

Diane Maclagan, Josephine Yu

Published 2019-08-16Version 1

We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0is (d-l)-connected through codimension one, where l is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity result for skeleta of rational polytopes. We also prove a tropical analogue of the Bertini Theorem: the intersection of the tropicalization of an irreducible variety with a generic hyperplane is again the tropicalization of an irreducible variety.