Birational map

Definition

A birational map is a rational map between algebraic varieties that is an isomorphism outside of a lower-dimensional subvariety. This concept plays a crucial role in understanding the relationships between varieties and determining when two varieties can be considered essentially the same from a geometric perspective. It connects directly to birational equivalence, where two varieties are considered equivalent if there exists a birational map between them.

5 Must Know Facts For Your Next Test

1. Birational maps allow us to understand how different varieties can be related, even if they appear different at first glance.
2. The domain and codomain of a birational map may have singularities or points where the map is not defined, but it is still considered a valid birational relationship.
3. Birational maps preserve certain geometric properties like dimension, allowing mathematicians to classify varieties based on these maps.
4. They are crucial for the minimal model program, where the goal is to simplify algebraic varieties through sequences of birational transformations.
5. Two varieties that are birationally equivalent might not be isomorphic as varieties; they may have different structures but can still be connected through birational maps.

Review Questions

• How does a birational map relate to the concepts of rational maps and algebraic varieties?
  • A birational map is built upon the foundation of rational maps, serving as a bridge between algebraic varieties. While rational maps can be thought of as functions defined by polynomial ratios, birational maps specifically refer to those that are isomorphisms outside of lower-dimensional subvarieties. Understanding this relationship helps clarify how different varieties can relate to each other even if they are not identical.
• Discuss the implications of birational equivalence on the classification of algebraic varieties.
  • Birational equivalence has significant implications for classifying algebraic varieties because it allows mathematicians to group varieties based on their essential geometric characteristics rather than their explicit structure. Two varieties that are birationally equivalent can exhibit similar properties despite differences in their forms. This classification helps streamline complex relationships and highlights underlying similarities in geometry.
• Evaluate the role of birational maps in the minimal model program and its significance in algebraic geometry.
  • Birational maps play a pivotal role in the minimal model program by facilitating transformations aimed at simplifying algebraic varieties into 'minimal' forms. This program seeks to construct models that possess desirable properties while preserving the essential features of the original variety. The significance lies in its ability to provide insights into the structure and classification of varieties, guiding researchers toward understanding complex geometric relationships and achieving canonical forms in algebraic geometry.

Related terms

Rational map:

A rational map is a function defined by ratios of polynomial functions between two varieties, which may not be defined everywhere.

Algebraic variety:

An algebraic variety is a fundamental object in algebraic geometry, representing the solution set of one or more polynomial equations.

Birational equivalence: Two varieties are birationally equivalent if there exists a birational map between them, indicating they share many important geometric properties.