Tropical compactification

From HandWiki

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 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

References

  1. Tevelev, Jenia (2007-08-07). "Compactifications of subvarieties of tori" (in en). American Journal of Mathematics 129 (4): 1087–1104. doi:10.1353/ajm.2007.0029. ISSN 1080-6377. https://muse.jhu.edu/article/218981/summary. 
  2. Brugallé, Erwan; Shaw, Kristin (2014). "A Bit of Tropical Geometry". The American Mathematical Monthly 121 (7): 563–589. doi:10.4169/amer.math.monthly.121.07.563. 
  • 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. Bibcode2014arXiv1410.2837C.