Tropical compactification

In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraic torus, introduced by Jenia Tevelev. 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 $T$ and a toric variety $\mathbb {P}$, the compactification ${\bar {X}}$ is tropical when the map

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

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

• Cavalieri, Renzo; Markwig, Hannah; Ranganathan, Dhruv (2017). "Tropical compactification and the Gromov–Witten theory of $\mathbb {P} ^{1}$". Selecta Mathematica. 23: 1027–1060. arXiv:1410.2837. Bibcode:2014arXiv1410.2837C. doi:10.1007/s00029-016-0265-7.