Complete Toric Variety

Definition

A complete toric variety is a specific type of toric variety that is projective and can be realized as a compactification of a torus. It is defined by the geometry of a fan, where each cone corresponds to an affine chart of the variety. This relationship connects it to important concepts like polyhedral geometry, combinatorial data, and algebraic varieties, making it a fundamental object in algebraic geometry.

5 Must Know Facts For Your Next Test

1. Complete toric varieties are projective, which means they can be represented as closed subsets in projective space, ensuring they are compact.
2. The construction of complete toric varieties relies heavily on the properties of fans, allowing for explicit descriptions through combinatorial geometry.
3. Each complete toric variety has an associated polytope, and the vertices of this polytope correspond to the torus-invariant subvarieties.
4. The study of complete toric varieties provides insights into intersection theory, allowing mathematicians to analyze the relationships between different subvarieties.
5. Complete toric varieties have applications in areas such as mirror symmetry and algebraic statistics, demonstrating their relevance beyond pure algebraic geometry.

Review Questions

• How does the concept of fans relate to the construction of complete toric varieties?
  • Fans are crucial in defining complete toric varieties as they provide the combinatorial structure necessary for organizing the cones that correspond to affine charts. Each cone in a fan describes how the local affine pieces fit together to create the global structure of the variety. Thus, understanding fans allows us to grasp how complete toric varieties emerge from simple geometrical shapes in higher-dimensional spaces.
• Discuss the implications of complete toric varieties being projective for their geometric properties.
  • Being projective implies that complete toric varieties are compact and can be studied using tools from projective geometry. This compactness guarantees that they exhibit properties like finiteness of certain invariants and allows for the application of intersection theory. The projective nature also ensures that these varieties have well-defined dimensions and can be embedded into projective spaces, linking them with classical algebraic geometry.
• Evaluate the significance of complete toric varieties in contemporary algebraic geometry research.
  • Complete toric varieties play a pivotal role in modern algebraic geometry due to their combinatorial nature and connection to various mathematical fields. They serve as testing grounds for ideas in mirror symmetry and facilitate computations in algebraic statistics. Furthermore, their structure allows researchers to derive properties about more complex algebraic varieties, making them fundamental objects for advancing both theoretical knowledge and practical applications.

Related terms

Toric Variety: A toric variety is an algebraic variety that is defined by combinatorial data associated with a fan, which organizes the way cones in a vector space interact.

Fan: A fan is a collection of cones in a vector space that satisfy certain intersection properties, serving as a combinatorial object that defines toric varieties.

Projective Variety: A projective variety is an algebraic variety that can be embedded in projective space, which means it is compact and has a well-defined geometric structure.