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 $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ 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.Comment: 36 pages, final version to appear in Duk

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

“…An alternative approach to the tropicalization of subvarieties of toric varieties, that is similar in spirit to our construction, has been developed by Giansiracusa and Giansiracusa in . Let be a free finitely generated abelian group and consider a rational polyhedral fan in .…”

“…Then defines a toric variety over the field with one element and the associated sharp monoidal space turns out to be a toric Kato fan. Instead of ‘analytifying’ , by considering the ‐valued points of , the authors of work in the category of semiring scheme and consider the base change , where denotes the semi‐ring of tropical numbers.…”

Syzygies over the polytope semiring

“…The Giansiracusa brothers study the notion of Hilbert polynomial of a tropical variety. For the toric ideal associated to the root lattice , their Hilbert polynomial is different from the polynomial underlying the Newton–Hilbert series.…”

“…For the toric ideal associated to the root lattice , their Hilbert polynomial is different from the polynomial underlying the Newton–Hilbert series. In fact, the tropical Hilbert polynomial defined in coincides with the Hilbert polynomial of the underlying ideal.…”

Toward homological characterization of semirings by e-injective semimodules

“…varieties [43] and [9], several mathematicians have studied from different points of view semiring schemes, in particular, in [21], sheaves and homological methods on semiring schemes have been considered.…”