Toric varieties are algebraic spaces with torus actions, bridging algebra and geometry. They're built from fans - collections of cones in lattices - and their properties are determined by the 's structure.
Fans give us a powerful tool to study toric varieties. We can analyze smoothness, completeness, and divisors just by looking at the fan. This connection between geometry and combinatorics is what makes toric varieties so useful and interesting.
Toric Varieties and Fans
Definition and Relationship to Fans
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A is an algebraic variety containing an algebraic torus as a dense open subset, allowing the torus action on itself to extend to an action on the entire variety
Toric varieties are constructed from combinatorial data called a fan, consisting of a collection of strongly convex rational polyhedral cones in a lattice
The combinatorial structure of the fan determines the torus action on the corresponding toric variety
The orbit-cone correspondence relates the orbits of the torus action on a toric variety to the cones in the associated fan
Toric varieties are normal algebraic varieties
Properties of Toric Varieties
Toric varieties are normal algebraic varieties
The dimension of a toric variety XΣ equals the dimension of the lattice N
XΣ is smooth if and only if each cone in Σ is generated by a subset of a basis of the lattice N
For example, the projective space Pn is a smooth toric variety
XΣ is complete (proper over C) if and only if the support of Σ is the entire lattice N
Complete toric varieties include projective spaces and weighted projective spaces
Constructing Toric Varieties
Construction from Fan Data
A fan Σ in a lattice N determines a toric variety XΣ
Each cone σ in the fan Σ corresponds to an Uσ, which is an open subset of XΣ
The affine toric variety Uσ is constructed as the spectrum of the semigroup algebra C[σ∨∩M], where M is the dual lattice of N and σ∨ is the dual cone of σ
The toric variety XΣ is obtained by gluing the affine toric varieties Uσ for all cones σ in Σ along their common open subsets
The gluing maps are determined by the inclusion relations among the cones in the fan Σ
Torus Action on Toric Varieties
The torus action on XΣ is induced by the natural action of the torus on each affine toric variety Uσ
The torus action on a toric variety is determined by the combinatorial structure of the corresponding fan
The orbit-cone correspondence relates the orbits of the torus action to the cones in the fan
Geometric Properties of Toric Varieties
Singularities and Smoothness
The singularities of XΣ correspond to the non-simplicial cones in Σ
A cone is simplicial if it is generated by linearly independent vectors
XΣ is smooth if and only if each cone in Σ is generated by a subset of a basis of the lattice N
Smooth toric varieties include projective spaces and products of projective spaces
Divisors and Intersection Theory
The divisor class group of XΣ can be computed from the combinatorial data of the fan Σ
Torus-invariant divisors on XΣ correspond to piecewise linear functions on the fan Σ
The intersection theory on XΣ is determined by the fan structure and can be computed combinatorially
The intersection numbers of torus-invariant divisors can be calculated using the fan data
Classifying Toric Varieties
Types of Toric Varieties
Affine toric varieties correspond to a single cone in the lattice N
The affine space An is an example of an affine toric variety
Projective toric varieties arise from fans that are the normal fans of lattice polytopes
Projective spaces Pn and products of projective spaces are projective toric varieties
Weighted projective spaces are toric varieties corresponding to fans with a single cone of dimension equal to the lattice rank
Weighted projective spaces generalize projective spaces by allowing different weights for the coordinates
Special Classes of Toric Varieties
Fano toric varieties are characterized by fans in which each cone is generated by a subset of a basis of the lattice, and the primitive generators of the rays span the lattice over Z
Fano toric varieties have ample anticanonical divisor and are used in mirror symmetry
Toric varieties with a torus-invariant point correspond to fans with a cone of maximal dimension
The affine space An and weighted projective spaces have torus-invariant points