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Equations of tropical varieties / Jeffrey Giansiracusa; Noah Giansiracusa

Duke Mathematical Journal, Volume: 165, Issue: 18, Pages: 3379 - 3433

Swansea University Author: Giansiracusa, Jeffrey

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 =(ℝ∪{−∞},max,+) by realizing them as solution sets to explicit systems of tropical equation...

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Published in: Duke Mathematical Journal
ISSN: 0012-7094
Published: 2016
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa26475
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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 =(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 -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 -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.
College: College of Science
Issue: 18
Start Page: 3379
End Page: 3433