Regular Map

Definition

A regular map is a smooth, continuous function between two algebraic varieties that preserves the structure of the varieties. It serves as a crucial tool in algebraic geometry to understand how different varieties relate to one another through morphisms, allowing for the study of their geometric and algebraic properties. Regular maps can be thought of as the 'morphisms' that facilitate the transition from one algebraic structure to another, playing a fundamental role in the classification of varieties and understanding their embeddings.

5 Must Know Facts For Your Next Test

1. Regular maps are defined using polynomial functions, meaning they can be expressed in terms of polynomials in the coordinates of the varieties involved.
2. They must be defined on an open set in the domain variety, ensuring that the function behaves well locally before extending it globally.
3. If a regular map is an isomorphism, it indicates that the two varieties are essentially the same from a structural viewpoint.
4. Regular maps play a key role in establishing connections between affine and projective varieties, allowing us to study their properties more comprehensively.
5. The composition of two regular maps is also a regular map, preserving the smoothness and continuity necessary for various applications in geometry.

Review Questions

• How do regular maps facilitate the understanding of relationships between different algebraic varieties?
  • Regular maps are crucial because they allow us to construct morphisms between algebraic varieties, preserving their underlying structure. By studying these maps, we can see how one variety can transform into another while retaining important properties. This relationship helps classify varieties and understand their embeddings, giving insight into their geometric characteristics and interconnections.
• In what ways do regular maps differ from isomorphisms, and why is this distinction important in algebraic geometry?
  • Regular maps can be any smooth functions between varieties, whereas isomorphisms are specific types of regular maps that establish a bijective relationship. This distinction is significant because while all isomorphisms are regular maps, not all regular maps are isomorphisms. Understanding this difference helps classify varieties more accurately and reveals whether they can be considered identical in structure or merely related through a more general mapping.
• Evaluate the impact of regular maps on the study of affine versus projective varieties, highlighting specific examples where applicable.
  • Regular maps play an essential role in bridging the gap between affine and projective varieties. For example, when dealing with an affine variety defined by polynomials, we can extend it to projective space using a regular map. This process allows us to explore properties such as compactness and intersection theory in projective settings while starting from more manageable affine structures. Such evaluations deepen our understanding of how different types of varieties relate within algebraic geometry.

Related terms

Morphisms:

Mappings between algebraic varieties that preserve their structure and relationships, fundamental in studying algebraic geometry.

Isomorphism:

A special type of morphism that establishes a bijective correspondence between two varieties, implying they are structurally identical.

Embedding: A regular map that allows one variety to be viewed as a subset of another variety while preserving its structure.