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
Famous quotes containing the words variety and/or fan:
“Any language is necessarily a finite system applied with different degrees of creativity to an infinite variety of situations, and most of the words and phrases we use are “prefabricated” in the sense that we don’t coin new ones every time we speak.”
—David Lodge (b. 1935)
“A matchmaker after a wedding is like a fan after autumn.”
—Chinese proverb.