Georgia Institute of Technology College of Sciences

I'll discuss joint work with J.H. Giansiracusa (Swansea) in which we study scheme theory over the tropical semiring T, using the notion of semiring schemes provided by Toen-Vaquie, Durov, or Lorscheid. We define tropical hypersurfaces in this setting and a tropicalization functor that sends closed subschemes of a toric variety over a field with non-archimedean valuation to closed subschemes of the corresponding toric variety over T. Upon passing to the set of T-valued points this yields Payne's extended tropicalization functor. We prove that the Hilbert polynomial of any projective subscheme is preserved by our tropicalization functor, so the scheme-theoretic foundations developed here reveal a hidden flatness in the degeneration sending a variety to its tropical skeleton.