## Duke Mathematical Journal

### 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.

#### Article information

Source
Duke Math. J., Volume 165, Number 18 (2016), 3379-3433.

Dates
Revised: 7 January 2016
First available in Project Euclid: 22 August 2016

https://projecteuclid.org/euclid.dmj/1471873079

Digital Object Identifier
doi:10.1215/00127094-3645544

Mathematical Reviews number (MathSciNet)
MR3577368

Zentralblatt MATH identifier
1342.14056

#### Citation

Giansiracusa, Jeffrey; Giansiracusa, Noah. Equations of tropical varieties. Duke Math. J. 165 (2016), no. 18, 3379--3433. doi:10.1215/00127094-3645544. https://projecteuclid.org/euclid.dmj/1471873079

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