Equations of tropical varieties

Abstract

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T}=(\mathbb{R}\cup \{-\infty \},\max ,+)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.

36 pages, final version to appear in Duke

References in corpus (4)

• The tropicalization of the moduli space of curves
• The Bogomolov conjecture for totally degenerate abelian varieties
• Picard Groups and Class Groups of Monoid Schemes
• Tropicalization of classical moduli spaces

Cited by in corpus (9)

• Functorial tropicalization of logarithmic schemes: The case of constant coefficients
• A moduli stack of tropical curves
• Tropical Ideals
• Tropical geometry over the tropical hyperfield
• A module-theoretic approach to matroids
• On congruence-semisimple semirings and the $K_0$-group characterization of ultramatricial algebras over semifields
• Equivariant vector bundles on toric schemes over semirings
• Dynamical Tropicalisation
• Geometry of tropical extensions of hyperfields