Birational equivalence

Definition

Birational equivalence is a relationship between algebraic varieties where two varieties are considered equivalent if they can be connected by rational maps that are inverses of each other on dense open subsets. This concept is fundamental in understanding how varieties can share similar properties, even if they are not isomorphic as schemes. Birational equivalence often arises in the study of projective varieties and their properties, where it helps in classifying varieties based on their geometric features.

5 Must Know Facts For Your Next Test

1. Two varieties are birationally equivalent if there exist rational maps between them that are defined on dense open subsets.
2. Birational equivalence is a weaker condition than isomorphism; it allows for two varieties to be considered equivalent without being structurally identical.
3. The notion of birational equivalence plays a crucial role in the Minimal Model Program, which seeks to classify algebraic varieties.
4. For smooth projective varieties, birational equivalence can often be tested using intersection theory and characteristics of their divisors.
5. When studying birational transformations, it’s important to consider the blow-up and contraction processes that help to relate different varieties.

Review Questions

• How does birational equivalence differ from isomorphism in the context of algebraic varieties?
  • Birational equivalence and isomorphism both describe relationships between algebraic varieties, but they differ in terms of strength. Isomorphism indicates a direct, structure-preserving correspondence between two varieties, meaning they can be transformed into each other without losing any properties. In contrast, birational equivalence allows for more flexibility; two varieties can be considered equivalent if they have rational maps connecting them, even if they are not directly isomorphic. This means that birationally equivalent varieties may have different structures but still share significant geometric characteristics.
• In what ways does birational equivalence influence the classification of projective varieties?
  • Birational equivalence significantly influences the classification of projective varieties by allowing mathematicians to group together varieties that share similar geometric properties despite potentially differing in structure. This classification can reveal important information about their behavior under various operations like blow-ups or contractions. Furthermore, understanding birational equivalence facilitates the development of theories like the Minimal Model Program, which seeks to classify varieties by simplifying them while preserving their essential features through birational transformations.
• Evaluate the importance of rational maps in establishing birational equivalence between two algebraic varieties.
  • Rational maps are essential for establishing birational equivalence because they provide the means through which two algebraic varieties can be connected. By examining the existence of rational maps that act as inverses on dense open subsets, mathematicians can identify whether two varieties share similar geometric properties. This connection is particularly relevant in cases where varieties cannot be directly compared via isomorphisms. The role of rational maps thus becomes central to understanding and utilizing birational equivalence in various algebraic geometry contexts, including classification and transformation processes.

Related terms

Rational Maps: Rational maps are functions between varieties that are defined by ratios of polynomial functions, which may be undefined on some subvarieties.

Isomorphism:

An isomorphism is a bijective morphism between varieties that establishes a one-to-one correspondence preserving the structure of the varieties.

Projective Space:

Projective space is a space of lines through the origin in a vector space, which serves as a natural setting for projective varieties.