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Equations of tropical varieties

Jeffrey Giansiracusa Orcid Logo, Noah Giansiracusa

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

Swansea University Author: Jeffrey Giansiracusa Orcid Logo

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DOI (Published version): 10.1215/00127094-3645544

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: Duke University Press 2016
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URI: https://cronfa.swan.ac.uk/Record/cronfa26475
first_indexed 2016-02-22T12:59:43Z
last_indexed 2021-01-07T03:42:04Z
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spelling 2021-01-06T16:34:28.3349318 v2 26475 2016-02-19 Equations of tropical varieties 03c4f93e1b94af60eb0c18c892b0c1d9 0000-0003-4252-0058 Jeffrey Giansiracusa Jeffrey Giansiracusa true false 2016-02-19 MACS 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. Journal Article Duke Mathematical Journal 165 18 3379 3433 Duke University Press 0012-7094 1 12 2016 2016-12-01 10.1215/00127094-3645544 http://dx.doi.org/10.1215/00127094-3645544 Pre-print version available via http://arxiv.org/abs/1308.0042 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2021-01-06T16:34:28.3349318 2016-02-19T21:06:09.9277727 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Jeffrey Giansiracusa 0000-0003-4252-0058 1 Noah Giansiracusa 2 0026475-19022016210956.pdf JG-NG-tropical.pdf 2016-02-19T21:09:56.6600000 Output 317851 application/pdf Accepted Manuscript true 2016-02-19T00:00:00.0000000 true
title Equations of tropical varieties
spellingShingle Equations of tropical varieties
Jeffrey Giansiracusa
title_short Equations of tropical varieties
title_full Equations of tropical varieties
title_fullStr Equations of tropical varieties
title_full_unstemmed Equations of tropical varieties
title_sort Equations of tropical varieties
author_id_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9
author_id_fullname_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9_***_Jeffrey Giansiracusa
author Jeffrey Giansiracusa
author2 Jeffrey Giansiracusa
Noah Giansiracusa
format Journal article
container_title Duke Mathematical Journal
container_volume 165
container_issue 18
container_start_page 3379
publishDate 2016
institution Swansea University
issn 0012-7094
doi_str_mv 10.1215/00127094-3645544
publisher Duke University Press
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
url http://dx.doi.org/10.1215/00127094-3645544
document_store_str 1
active_str 0
description 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.
published_date 2016-12-01T12:54:29Z
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score 11.048042

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