{{Short description|Mathematical concept}} In algebraic geometry, a '''tropical compactification''' is a compactification (projective completion) of a subvariety of an 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|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|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 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 and <math>\bar{X}</math> is proper.
== See also == *Tropical geometry *GIT quotient *Chow quotient *Toroidal embedding
== References == [[File:Markwig bertram cavalieri.jpg|thumb|''From left:'' Hannah Markwig, Aaron Bertram, and Renzo Cavalieri, 2012 at the MFO]] {{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–Witten theory of <math>\mathbb{P}^1</math>|arxiv=1410.2837|journal=Selecta Mathematica|volume=23|pages=1027–1060|doi=10.1007/s00029-016-0265-7 |bibcode=2014arXiv1410.2837C}}
Compactification
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