Toric variety

In algebraic geometry, a toric variety or torus embedding is a special type of algebraic variety. It includes something called an algebraic torus. The torus helps shape the whole variety.

Toric varieties are very useful in algebraic geometry. They often help test new ideas and rules.

The shape of a toric variety depends on the combinatorics of something called its associated fan. This makes many calculations easier. For some special toric varieties, this information can also be shown using a shape called a convex polytope. This links the study of toric varieties to a part of math called convex geometry.

Some well-known toric varieties include affine space, projective spaces, products of projective spaces, and certain bundles on projective space. These examples show how useful toric varieties are in math.

Definition

A toric variety is a special kind of shape in a part of math called algebraic geometry. It includes something named an algebraic torus, which is a special geometric object. This torus fits inside the toric variety in a perfect way. One important feature is how the torus moves, and this movement can affect the whole variety.

Toric varieties from tori

Toric varieties are special shapes studied in a part of math called algebraic geometry. They start with something called an algebraic torus. This is a geometric object made from equations.

By picking certain points and using them to make maps, we can build new shapes called affine varieties. If these points fully represent the original torus, the new shape is called a torus embedding.

We can also create shapes in projective space. This means we add points at infinity to get a complete shape. These shapes can be studied by looking at special lines inside them. By organizing these lines based on where they end, we get a structure called a lattice fan. This helps us understand the shape’s geometry.

The toric variety of a fan

Toric varieties are special shapes in algebraic geometry. They contain a smaller shape called an algebraic torus. These varieties help mathematicians test ideas and solve problems more easily.

The shape of a toric variety is decided by something called a "fan." This fan is made of cones. Cones are pointed shapes that start at the origin. By studying these cones and how they fit together, mathematicians can understand the whole toric variety. This makes calculations and proofs simpler.

Classification of smooth complete toric varieties

The classification of smooth complete toric varieties depends on their dimension and the number of rays in their associated fan. For varieties with a Picard number of 1, the only example is the complex projective space, which relates to the fan of a simplex.

For Picard number 2, all such varieties were classified by P. Kleinschmidt, and for Picard number 3, Victor V. Batyrev classified them. Classifications for higher Picard numbers are still unknown. Smooth toric surfaces are well-understood and can be described using polygons with special properties.

Resolution of singularities

Every toric variety can be made smoother by using another toric variety. This is done by breaking down the big parts of its shape into smaller, smoother parts. This helps make calculations easier and the geometry easier to understand.

Main article: resolution of singularities

Relation to mirror symmetry

The idea of toric varieties helps us understand something called mirror symmetry. By looking at certain shapes in a special way, we can create mirror versions of these shapes. This makes it easier to study and compare these mirror shapes.

Main article: mirror symmetry
Further information: combinatorial construction