In algebraic geometry, a toric variety or torus embedding 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.
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 polytope, which creates a powerful connection of the subject with convex geometry.
By taking the Zariski closure of the image of such a map, one obtains an affine variety.
generates the character lattice, this variety is a torus embedding.
Given a projective toric variety, observe that we may probe its geometry by one-parameter subgroups.
Each one parameter subgroup, determined by a point in the lattice, dual to the character lattice, is a punctured curve inside the projective toric variety.
Since the variety is compact, this punctured curve has a unique limit point.
Thus, by partitioning the one-parameter subgroup lattice by the limit points of punctured curves, we obtain a lattice fan, a collection of polyhedral rational cones.
The cones of highest dimension correspond precisely to the torus fixed points, the limits of these punctured curves.
is a finite-rank free abelian group, for instance the lattice
) with apex at the origin, generated by a finite number of vectors of
[further explanation needed] A fan is a collection of cones closed under taking intersections and faces.
The toric variety of a fan of strongly convex rational cones is given by taking the affine toric varieties of its cones and gluing them together by identifying
The toric variety constructed from a fan is necessarily normal.
Conversely, every toric variety has an associated fan of strongly convex rational cones.
This correspondence is called the fundamental theorem for toric geometry, and it gives a one-to-one correspondence between normal toric varieties and fans of strongly convex rational cones.
Moreover, a toric variety is smooth, or nonsingular, if every cone in its fan can be generated by a subset of a basis for the free abelian group
is the cone generated by the outer normals of the facets containing
It is well known that projective toric varieties are the ones coming from the normal fans of rational polytopes.
It may be represented by three complex coordinates satisfying where the sum has been chosen to account for the real rescaling part of the projective map, and the coordinates must be moreover identified by the following
action: The approach of toric geometry is to write The coordinates
However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at
The precise orientation of the circles within the torus is usually depicted by the slope of the line intervals (the sides of the triangle, in this case).
Note that this construction is related to symplectic geometry as the map
Cartier divisors is equivalent to that of smooth complete fans of dimension
Note that it is actually the rank of the Picard group of the toric variety associated to
Smooth toric surfaces are easily characterized, they all are projective and come from the normal fan of polygons such that at each vertex, the two incident edges are spanned by two vectors that form a basis of
The idea of toric varieties is useful for mirror symmetry because an interpretation of certain data of a fan as data of a polytope leads to a combinatorial construction of mirror manifolds.