Toric variety

In algebraic geometry, a toric variety or torus embedding is a special kind of algebraic variety that includes an algebraic torus. This torus's group action spreads out to affect the entire variety. Toric varieties are very important in algebraic geometry because they often serve as useful examples to test new ideas and theorems.

The shape and structure of a toric variety are completely decided by the combinatorics of something called its associated fan. This makes many calculations much easier to handle. For some special types of toric varieties, this information can also be shown using a shape called a convex polytope. This creates a strong link between the study of toric varieties and the area of mathematics known as convex geometry.

Some well-known examples of toric varieties include affine space, projective spaces, products of projective spaces, and certain bundles that sit on top of projective space. These examples show how useful and widespread toric varieties are in the world of mathematics.

Definition

A toric variety is a special kind of shape studied in a branch of math called algebraic geometry. It contains something called an algebraic torus, which is like a geometric object, and this torus fits perfectly inside the variety. One key feature is that the way the torus acts or moves on itself can be extended to affect the whole variety. Some experts also add that the toric variety should be normal, which is another technical condition they like to include.

Toric varieties from tori

Toric varieties are special types of shapes studied in a branch of math called algebraic geometry. They begin with something called an algebraic torus, which is like a geometric object formed from equations. By choosing certain points and using them to create maps, we can build new shapes called affine varieties. When these points fully represent the original torus, the new shape is called a torus embedding.

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

The toric variety of a fan

Toric varieties are special kinds of shapes in algebraic geometry that contain a smaller shape called an algebraic torus. These varieties are important because they help mathematicians test ideas and solve problems more easily.

The geometry of a toric variety is completely decided by a structure called a "fan." This fan is made up of cones, which are like 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 much simpler than they would otherwise be.

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 is linked 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, with both sets being projective. Classifications for higher Picard numbers remain unknown. Smooth toric surfaces are well-understood and can be described using polygons with specific 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 and their properties in a special way, we can build mirror versions of these shapes using simple, organized rules. This makes it easier to study and compare these mirror shapes.

Main article: mirror symmetry
Further information: combinatorial construction