Faithful tropicalization of hypertoric varieties

A hypertoric variety is a “hyperkohler analogue” of a toric variety. Each hypertoric variety comes equipped with an embedding into a toric variety, called the Lawrence toric variety, and hence has a natural tropicalization. We explicitly describe the polyhedral structure of this tropicalization. Using a recent result of Gubler, Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization map.