Rational Map

Definition

A rational map is a function between varieties that is defined by ratios of polynomials. Essentially, it allows us to describe how two algebraic varieties are related to each other through this kind of function, especially when both varieties are not necessarily isomorphic. This concept is important when examining how varieties can be embedded within one another and when considering birational equivalence, where the rational map may only be defined on a dense open subset.

5 Must Know Facts For Your Next Test

1. A rational map can be seen as a generalization of a morphism, particularly when the morphism is not defined everywhere on the varieties involved.
2. Rational maps are often expressed in terms of coordinates, making them useful for computing properties of varieties using algebraic techniques.
3. The notion of birational equivalence hinges on the existence of rational maps that relate different varieties, allowing for flexibility in understanding their structure.
4. If two varieties are birationally equivalent, there exist rational maps in both directions, meaning you can go from one variety to another and back with these maps.
5. In many cases, the indeterminacy of a rational map arises at certain points where the polynomials defining it may not be well-defined, leading to interesting geometric interpretations.

Review Questions

• How does a rational map differ from a morphism when relating two algebraic varieties?
  • A rational map differs from a morphism primarily in that it is not required to be defined everywhere on the varieties. While a morphism is a well-defined function at all points of the varieties involved, a rational map may have points of indeterminacy where it is not defined. This means that while all morphisms can be seen as rational maps, not all rational maps qualify as morphisms.
• Discuss how the concept of birational equivalence relies on rational maps and what implications this has for understanding geometric structures.
  • Birational equivalence relies on the existence of rational maps connecting two varieties, suggesting that despite possible differences in their structures, they can share significant similarities. This relationship allows mathematicians to study complex varieties by comparing simpler or more well-understood ones. Essentially, if two varieties are birationally equivalent through rational maps, it implies they have the same function field and can thus be treated similarly in certain contexts.
• Evaluate the significance of indeterminacy in rational maps and its impact on understanding the geometry of varieties.
  • Indeterminacy in rational maps signifies points where the map fails to be well-defined, which raises important questions about the geometry of the varieties involved. Understanding where these indeterminate points lie helps reveal deeper properties about the relationships between varieties. For example, analyzing indeterminacy can lead to insights about how curves intersect or how surfaces might behave under certain conditions, ultimately affecting how we conceptualize their geometric nature.

Related terms

Isomorphism: A bijective morphism between two varieties that preserves the structure, indicating that the two varieties are essentially the same in terms of their geometric properties.

Birational Equivalence: A relationship between two varieties where they can be connected via rational maps, meaning they have rational functions that define a correspondence between them, though they may not be isomorphic.

Polynomial Ring: An algebraic structure formed from polynomials, which serves as the domain for defining rational maps and their properties.