Pullback

Definition

In algebraic geometry, a pullback is a way to transfer functions or cohomology classes from one space to another via a morphism or rational map. It allows us to understand how properties and structures in one geometric setting relate to another, particularly when considering Chow rings and intersection theory or analyzing the behavior of maps between different varieties.

5 Must Know Facts For Your Next Test

1. The pullback operation can be applied to both functions and differential forms, allowing the translation of information between varieties.
2. In the context of Chow rings, the pullback is essential for understanding how classes in one variety relate to those in another under a given morphism.
3. Pullbacks preserve intersections, meaning that the intersection of pulled-back classes corresponds to the intersection in the original varieties.
4. When dealing with rational maps, pullbacks can often be more complicated due to potential indeterminacies or singularities that may arise.
5. The pullback of divisors under a morphism can help analyze the behavior of maps between varieties, especially when investigating their ramification and degree.

Review Questions

• How does the pullback operation facilitate the understanding of morphisms between algebraic varieties?
  • The pullback operation allows us to transfer functions or classes from one variety to another via a morphism, which helps in analyzing how geometric properties are preserved or transformed. This understanding is crucial for studying relationships between different varieties and helps us investigate their structure through algebraic invariants. By applying the pullback, we can observe how specific features like divisors or cohomology classes behave under the mapping.
• Discuss how pullbacks interact with Chow rings and their role in intersection theory.
  • Pullbacks are integral to working with Chow rings because they enable us to relate the cycles on different varieties through morphisms. When we take the pullback of a class from one Chow ring, it gives us insights into how intersections behave when mapped to another variety. This interaction allows us to utilize intersection theory effectively by preserving essential information about intersections and dimensionality across different spaces.
• Evaluate the significance of pullbacks in both morphisms and rational maps, considering their implications for singularities.
  • Pullbacks play a critical role in both morphisms and rational maps by enabling the transfer of algebraic information across varieties. However, when dealing with rational maps, pullbacks can lead to complications due to potential indeterminacies where the map is not defined. Understanding these intricacies is vital as it helps mathematicians manage singularities that may arise during this process, ultimately influencing how properties of the original varieties are represented in the target space.

Related terms

Morphisms: Structure-preserving maps between algebraic varieties that allow for the transfer of geometric and algebraic properties.

Cohomology: A mathematical tool used to study topological spaces and algebraic varieties, providing a way to classify and analyze their properties using algebraic techniques.

Chow Rings: Algebraic invariants that encode information about the cycles on a variety, playing a crucial role in intersection theory and the study of algebraic geometry.