Toric geometry
Encyclopedia
In algebraic geometry
, a toric variety or torus embedding is a normal variety containing an algebraic torus
as a dense subset, such that the action
of the torus on itself extends to the whole variety.
. 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.
This implies that every toric variety has a resolution of singularities
given by another toric variety, which can be constructed by subdividing the maximal cones into cones of nonsingular toric varieties.
The toric variety has a map to the polytope in the dual of N whose fibers are topological tori. For example, the complex projective plane CP2 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 U(1) action:
The approach of toric geometry is to write
The coordinates
are non-negative, and they parameterize a triangle because
that is,
The triangle is the toric base of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of
; the phase of 
can be chosen real and positive by the 
symmetry.
However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at
or 
or 
because the phase of 
becomes inconsequential, respectively.
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).
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a toric variety or torus embedding is a normal variety containing an algebraic torus
Algebraic torus
In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory...
as a dense subset, such that the action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of the torus on itself extends to the whole variety.
The toric variety of a fan
Suppose that N is a finite-rank free abelian groupFree abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
. 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
Basis (Universal Algebra)
In universal algebra a basis is a structure inside of some algebras, which are called free algebras. It generates all algebra elements from its own elements by the algebra operations in an independent manner...
for the free abelian group N.
Morphisms of toric varieties
Suppose that Δ1 and Δ2 are fans in lattices N1 and N2. If f is a linear map from N1 to N2 such that the image of every cone of Δ1 is contained in a cone of Δ2, then f induces a morphism f* between the corresponding toric varieties. This map f* is proper if and only if the map f maps |Δ1| onto |Δ2|, where |Δ| is the underlying space of a fan Δ given by the union of its cones.Resolution of singularities
A toric variety is nonsingular if its cones of maximal dimension are generated by a basis of the lattice.This implies that every toric variety has a resolution of singularities
Resolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V...
given by another toric variety, which can be constructed by subdividing the maximal cones into cones of nonsingular toric varieties.
The toric variety of a convex polytope
The fan of a rational convex polytope in N consists of the cones over its proper faces. The toric variety of the polytope is the toric variety of its fan. A variation of this construction is to take a rational polytope in the dual of N and take the toric variety of its polar set in N.The toric variety has a map to the polytope in the dual of N whose fibers are topological tori. For example, the complex projective plane CP2 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 U(1) action:
The approach of toric geometry is to write
The coordinates
are non-negative, and they parameterize a triangle becausethat is,
The triangle is the toric base of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of
; the phase of can be chosen real and positive by the symmetry.However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at
or or because the phase of becomes inconsequential, respectively.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).
External links
- Home page of D. A. Cox, with several lectures on toric varieties