# Higher Connectivity of Tropicalizations

## TL;DR

This paper establishes higher connectivity properties of tropicalizations of algebraic varieties and proves a tropical Bertini theorem, revealing new structural insights in tropical geometry.

## Contribution

It introduces a higher connectivity theorem for tropicalizations and a tropical analogue of Bertini's theorem, extending classical results to tropical geometry.

## Key findings

- Tropicalization of an irreducible variety is (d-l)-connected through codimension one.
- The intersection of a tropicalized variety with a generic hyperplane remains irreducible.
- Provides new tools for understanding the topology of tropical varieties.

## Abstract

We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0 is (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.