Target

Definition

In the context of rational maps between varieties, a target refers to the codomain or image of a rational map, which connects two algebraic varieties. The target is crucial in understanding how points from the domain (the source variety) map to points in the target variety, revealing the structure and properties of both varieties involved. Rational maps help identify how one variety can be represented or transformed in relation to another.

5 Must Know Facts For Your Next Test

1. The target of a rational map can sometimes be different from its original variety, especially when considering images under the map that may not cover the entire target.
2. Understanding the target is essential for studying morphisms and their properties, including birational equivalence.
3. The target can have various geometric structures and properties that influence how the mapping occurs between varieties.
4. In cases where a rational map is not defined everywhere, it is important to consider how this affects the relationship between the domain and the target.
5. When analyzing rational maps, one must often look at the fibers over points in the target to understand pre-images and their implications.

Review Questions

• How does understanding the target of a rational map enhance your comprehension of its geometric implications?
  • Understanding the target of a rational map is essential because it helps clarify how points from one variety relate to another. By examining the structure and properties of the target, one can determine how well the map preserves certain features or introduces new ones. This understanding allows us to analyze geometric transformations more effectively and to explore relationships such as birational equivalence between varieties.
• Discuss the significance of analyzing fibers over points in the target when studying rational maps.
  • Analyzing fibers over points in the target reveals important information about how many pre-images correspond to each point in the target variety. This analysis helps identify critical behaviors of the rational map, such as whether it is injective or surjective at certain points. Such investigations are vital for understanding not just individual mappings but also broader relationships and behaviors across various varieties.
• Evaluate how different structures within a target can impact the overall behavior of a rational map between varieties.
  • Different structures within a target can significantly influence how a rational map behaves by affecting aspects like continuity, singularities, and dimension. For instance, if a target variety has singular points or varying dimensions, this can lead to unexpected behavior in terms of mapping and image coverage. An evaluation of these structural differences helps mathematicians understand not only specific mappings but also contributes to larger theories regarding morphisms and their classifications across various algebraic contexts.

Related terms

Rational Map: A rational map is a function between two varieties that is defined by rational functions, which can be thought of as a way to encode the relationship between the two varieties.

Domain: The domain is the source variety from which points are mapped to another variety through a rational map.

Image: The image is the set of all points in the target variety that correspond to points from the domain via the rational map.