Tropical compactification explained
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
and a toric variety
, the compactification
is tropical when the map
\Phi:T x \bar{X}\toP, (t,x)\totx
is
faithfully flat and
is proper.
See also
References
- Cavalieri. Renzo. Markwig. Hannah. Hannah Markwig. Ranganathan. Dhruv. 2017. Tropical compactification and the Gromov - Witten theory of
. 1410.2837. Selecta Mathematica. 23. 1027 - 1060. 10.1007/s00029-016-0265-7 . 2014arXiv1410.2837C.
Notes and References
- Tevelev. Jenia. 2007-08-07. Compactifications of subvarieties of tori. American Journal of Mathematics. en. 129. 4. 1087–1104. math/0412329. 10.1353/ajm.2007.0029. 1080-6377.
- Brugallé. Erwan. Shaw. Kristin. 2014. A Bit of Tropical Geometry. 10.4169/amer.math.monthly.121.07.563. The American Mathematical Monthly. 121. 7. 563–589. 10.4169/amer.math.monthly.121.07.563. 1311.2360.
