# Tropical compactification

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{{Short description|Mathematical concept}}
In [algebraic geometry](/source/algebraic_geometry), a '''tropical compactification''' is a compactification ([projective completion](/source/projective_completion)) of a [subvariety](/source/algebraic_variety) of an [algebraic torus](/source/algebraic_torus), introduced by Jenia Tevelev.<ref>{{Cite journal|last=Tevelev|first=Jenia|date=2007-08-07|title=Compactifications of subvarieties of tori|url=https://muse.jhu.edu/article/218981/summary|journal=[American Journal of Mathematics](/source/American_Journal_of_Mathematics)|language=en|volume=129|issue=4|pages=1087–1104|arxiv=math/0412329|doi=10.1353/ajm.2007.0029|issn=1080-6377}}</ref><ref>{{Cite journal|last1=Brugallé|first1=Erwan|last2=Shaw|first2=Kristin|date=2014|title=A Bit of Tropical Geometry|jstor=10.4169/amer.math.monthly.121.07.563|journal=[The American Mathematical Monthly](/source/The_American_Mathematical_Monthly)|volume=121|issue=7|pages=563–589|doi=10.4169/amer.math.monthly.121.07.563|arxiv=1311.2360}}</ref> Given an algebraic torus and a connected closed subvariety of that torus, a compactification of the subvariety is defined as a closure of it in a [toric variety](/source/toric_variety) of the original torus. The concept of a tropical compactification arises when trying to make compactifications as "nice" as possible. For a torus <math>T</math> and a toric variety <math>\mathbb{P}</math>, the compactification <math>\bar{X}</math> is tropical when the map
:<math>\Phi: T \times \bar{X} \to \mathbb{P},\ (t,x) \to tx</math>
is [faithfully flat](/source/flat_morphism) and <math>\bar{X}</math> is proper.

== See also ==
*[Tropical geometry](/source/Tropical_geometry)
*[GIT quotient](/source/GIT_quotient)
*[Chow quotient](/source/Chow_quotient)
*[Toroidal embedding](/source/Toroidal_embedding)

== References ==
[[File:Markwig bertram cavalieri.jpg|thumb|''From left:'' Hannah Markwig, Aaron Bertram, and Renzo Cavalieri, 2012 at the [MFO](/source/Mathematical_Research_Institute_of_Oberwolfach)]]
{{Reflist}}

*{{Cite journal|last1=Cavalieri|first1=Renzo|last2=Markwig|first2=Hannah|author2-link = Hannah Markwig|last3=Ranganathan|first3=Dhruv|year=2017|title=Tropical compactification and the Gromov&ndash;Witten theory of <math>\mathbb{P}^1</math>|arxiv=1410.2837|journal=Selecta Mathematica|volume=23|pages=1027&ndash;1060|doi=10.1007/s00029-016-0265-7 |bibcode=2014arXiv1410.2837C}}

Compactification

{{Algebra-stub}}

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