1 December 2016 Equations of tropical varieties
Jeffrey Giansiracusa, Noah Giansiracusa
Duke Math. J. 165(18): 3379-3433 (1 December 2016). DOI: 10.1215/00127094-3645544


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 T=(R{},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 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 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.


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Jeffrey Giansiracusa. Noah Giansiracusa. "Equations of tropical varieties." Duke Math. J. 165 (18) 3379 - 3433, 1 December 2016. https://doi.org/10.1215/00127094-3645544


Received: 13 June 2014; Revised: 7 January 2016; Published: 1 December 2016
First available in Project Euclid: 22 August 2016

zbMATH: 1342.14056
MathSciNet: MR3577368
Digital Object Identifier: 10.1215/00127094-3645544

Primary: 14T05
Secondary: 14A20

Keywords: bend relations , Hilbert polynomial , max-plus algebra , Tropical geometry , tropical scheme , tropicalization

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 18 • 1 December 2016
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