Toric variety

In algebraic geometry, a toric variety or torus embedding is a kind of algebraic variety that contains an algebraic torus whose group action extends to the whole variety. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special but still quite general class of toric varieties, this information is also encoded in a convex polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space.

Definition

A precise definition is that a toric variety is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety.

Some authors also require it to be normal.

Toric varieties from tori

The original motivation to study toric varieties was to study torus embeddings. Given the algebraic torus

T

{\displaystyle T}

, the group of characters

hom

⁡

(

T

,

C

∗

)

{\displaystyle \hom(T,\mathbb {C} ^{*})}

forms a lattice. Given a collection of points

A

{\displaystyle {\mathcal {A}}}

, a subset of this lattice, each point determines a map to

C

∗

{\displaystyle \mathbb {C} ^{*}}

and thus the collection determines a map to

(

C

∗

)

|

A

|

.