9.1 Toric varieties and fans

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 fan'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 toric variety 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 $X_Σ$ equals the dimension of the lattice $N$
• $X_Σ$ 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 $\mathbb{P}^n$ is a smooth toric variety
• $X_Σ$ is complete (proper over $\mathbb{C}$) 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 $X_Σ$
• Each cone $σ$ in the fan $Σ$ corresponds to an affine toric variety $U_σ$, which is an open subset of $X_Σ$
  • The affine toric variety $U_σ$ is constructed as the spectrum of the semigroup algebra $\mathbb{C}[σ^∨ ∩ M]$, where $M$ is the dual lattice of $N$ and $σ^∨$ is the dual cone of $σ$
• The toric variety $X_Σ$ is obtained by gluing the affine toric varieties $U_σ$ 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 $X_Σ$ is induced by the natural action of the torus on each affine toric variety $U_σ$
• 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 $X_Σ$ correspond to the non-simplicial cones in $Σ$
  • A cone is simplicial if it is generated by linearly independent vectors
• $X_Σ$ 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 $X_Σ$ can be computed from the combinatorial data of the fan $Σ$
  • Torus-invariant divisors on $X_Σ$ correspond to piecewise linear functions on the fan $Σ$
• The intersection theory on $X_Σ$ 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 $\mathbb{A}^n$ is an example of an affine toric variety
• Projective toric varieties arise from fans that are the normal fans of lattice polytopes
  • Projective spaces $\mathbb{P}^n$ 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 $\mathbb{Z}$
  • 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 $\mathbb{A}^n$ and weighted projective spaces have torus-invariant points