Higher connectivity of tropicalizations

We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0 is $$(d-\ell )$$( d - ℓ ) -connected through codimension one, where $$\ell $$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.