In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraic torus, introduced by Jenia Tevelev.[1][2] Given an algebraic torus and a connected closed subvariety of that torus, a compatification of the subvariety is defined as a closure of it in a toric variety of the original torus. The concept of a tropical compatification arises when trying to make compactifications as "nice" as possible. For a torus T, a toric variety \( \mathbb {P} \) , the compatification \( {\bar {X}} \) is tropical when the map
\( {\displaystyle \Phi :T\times {\bar {X}}\to \mathbb {P} ,\ (t,x)\to tx} \)
is faithfully flat and \( {\bar {X}} \) is proper.
See also
Tropical geometry
GIT quotient
Chow quotient
Toroidal embedding
References
From left: Hannah Markwig, Aaron Bertram, and Renzo Cavalieri, 2012 at the MFO
Tevelev, Jenia (2007-08-07). "Compactifications of subvarieties of tori". American Journal of Mathematics. 129 (4): 1087–1104. arXiv:math/0412329. doi:10.1353/ajm.2007.0029. ISSN 1080-6377.
Brugallé, Erwan; Shaw, Kristin (2014). "A Bit of Tropical Geometry". The American Mathematical Monthly. 121 (7): 563–589. arXiv:1311.2360. doi:10.4169/amer.math.monthly.121.07.563. JSTOR 10.4169/amer.math.monthly.121.07.563.
Cavalieri, Renzo; Markwig, Hannah; Ranganathan, Dhruv (2017). "Tropical compactification and the Gromov–Witten theory of P 1 {\displaystyle \mathbb {P} ^{1}} \mathbb {P} ^{1}". Selecta Mathematica. 23: 1027–1060. arXiv:1410.2837. Bibcode:2014arXiv1410.2837C.
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