Tropical compactification

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 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 ${\displaystyle T}$ and a toric variety ${\displaystyle \mathbb {P} }$, the compactification ${\displaystyle {\bar {X}}}$ is tropical when the map

${\displaystyle \Phi :T\times {\bar {X}}\to \mathbb {P} ,\ (t,x)\to tx}$

is faithfully flat and ${\displaystyle {\bar {X}}}$ 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

1. 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.
2. 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 ${\displaystyle \mathbb {P} ^{1}}$". Selecta Mathematica. 23: 1027–1060. arXiv:1410.2837. Bibcode:2014arXiv1410.2837C. doi:10.1007/s00029-016-0265-7.