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.