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 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 [math]\displaystyle{ T }[/math], a toric variety [math]\displaystyle{ \mathbb{P} }[/math], the compatification [math]\displaystyle{ \bar{X} }[/math] is tropical when the map

[math]\displaystyle{ \Phi: T \times \bar{X} \to \mathbb{P},\ (t,x) \to tx }[/math]

is faithfully flat and [math]\displaystyle{ \bar{X} }[/math] is proper.

See also

• Tropical geometry
• GIT quotient
• Chow quotient
• Toroidal embedding

References

• Cavalieri, Renzo; Markwig, Hannah; Ranganathan, Dhruv (2017). "Tropical compactification and the Gromov–Witten theory of [math]\displaystyle{ \mathbb{P}^1 }[/math]". Selecta Mathematica 23: 1027–1060. Bibcode: 2014arXiv1410.2837C.