The Toric Variety of A Fan
Suppose that N is a finite-rank free abelian group. A strongly convex rational polyhedral cone in N is a convex cone (of the real vector space of N) with apex at the origin, generated by a finite number of vectors of N, that contains no line through the origin. These will be called "cones" for short.
For each cone σ its affine toric variety Uσ is the spectrum of the semigroup algebra of the dual cone.
A fan is a collection of cones closed under taking intersections and faces.
The toric variety of a fan is given by taking the affine toric varieties of its cones and glueing them together by identifying Uσ with an open subvariety of Uτ whenever σ is a face of τ. Conversely, every fan of strongly convex rational cones has an associated toric variety.
The fan associated with a toric variety condenses some important data about the variety. For example, a variety is smooth if every cone in its fan can be generated by a subset of a basis for the free abelian group N.
Read more about this topic: Toric Variety
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