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