Birational Equivalence

Definition

Birational equivalence is a concept in algebraic geometry where two varieties are considered equivalent if they can be related by rational maps that are inverses of each other on dense open subsets. This idea connects to regular functions and morphisms, as birational maps allow for understanding the behavior of functions across varieties. It's crucial for studying properties like dimension and classification, especially when dealing with surfaces and their geometric properties.

5 Must Know Facts For Your Next Test

1. Birational equivalence implies that two varieties share many properties, such as dimension and genus, even if they are not isomorphic.
2. In birational geometry, one often considers the behavior of rational maps rather than regular morphisms, as these can provide insights into the structure of the varieties involved.
3. Birational equivalence is a key tool in the classification of algebraic surfaces, allowing for the identification of different types based on their geometric properties.
4. The notion of birational equivalence extends beyond surfaces to higher-dimensional varieties, playing an important role in the study of Fano varieties and Calabi-Yau manifolds.
5. Understanding birational equivalence can lead to important results regarding the existence of certain divisors on algebraic varieties, as well as their intersection theory.

Review Questions

• How does birational equivalence relate to the concept of rational maps in algebraic geometry?
  • Birational equivalence relies heavily on rational maps, which connect two varieties by allowing functions to be defined except on lower-dimensional subsets. When two varieties are birationally equivalent, there exist rational maps that act as inverses on dense open sets, showing how they relate. This relationship emphasizes the importance of rationality in understanding the structure and properties of the varieties.
• Discuss the implications of birational equivalence for the classification of algebraic surfaces.
  • Birational equivalence plays a crucial role in classifying algebraic surfaces by providing a framework to analyze their geometric properties through rational maps. Surfaces that are birationally equivalent often share key invariants like dimension and Kodaira dimension, which helps mathematicians categorize them into classes such as Fano or ruled surfaces. This classification aids in understanding their deeper geometric features and relationships.
• Evaluate how birational equivalence affects the study of divisors and intersection theory on algebraic varieties.
  • Birational equivalence significantly impacts the study of divisors and intersection theory by allowing mathematicians to transfer information between varieties that may not be directly related. Since birationally equivalent varieties can exhibit similar behaviors concerning divisors, results derived from one variety can often be applied to another. This connection enriches intersection theory, providing insights into how divisors interact and behave across different geometrical contexts.

Related terms

Rational Map: A rational map is a function between varieties that is defined by polynomials except on a lower-dimensional subset.

Regular Function: A regular function is a function defined on an algebraic variety that can be expressed as a quotient of polynomial functions.

Algebraic Surface: An algebraic surface is a two-dimensional algebraic variety, which can be studied in terms of its geometric properties and classification.