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 $T$, a toric variety $\mathbb{\mathbb{P}}$, the compatification $\overline{X}$ is tropical when the map

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

is faithfully flat and $\overline{X}$ 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 ${\mathbb{\mathbb{P}}}^1$". Selecta Mathematica 23: 1027–1060. Bibcode: 2014arXiv1410.2837C.