Toric variety

Toric variety

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.


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.

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 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

|z_1|^2+|z_2|^2+|z_3|^2 = 1 , \,\!

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:

(z_1,z_2,z_3)\approx e^{i\phi} (z_1,z_2,z_3) . \,\!

The approach of toric geometry is to write

(x,y,z) = (|z_1|^2,|z_2|^2,|z_3|^2) . \,\!

The coordinates x,y,z are non-negative, and they parameterize a triangle because

x+y+z=1 ; \,\!

that is,

\quad z=1-x-y . \,\!

The triangle is the toric base of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of z1,z2; the phase of z3 can be chosen real and positive by the U(1) symmetry.

However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at x = 0 or y = 0 or z = 0 because the phase of z1,z2,z3 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

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