Morphism between curves

Definition

A morphism between curves is a map between two algebraic curves that preserves the structure of the curves, meaning it respects their geometric and algebraic properties. In algebraic geometry, these morphisms are crucial as they allow for the comparison of different curves and can reveal important relationships between them. A morphism can be thought of as a way to translate points from one curve to another while maintaining the underlying algebraic structure.

5 Must Know Facts For Your Next Test

1. Morphism between curves can be represented by polynomial functions in terms of their coordinates, which allows us to study their properties using algebraic expressions.
2. Morphism can be classified as either dominant or non-dominant; a dominant morphism is one where the image is dense in the target curve.
3. When two curves are connected by a morphism, it can provide insights into their geometric properties, such as genus or singularities.
4. The composition of two morphisms between curves results in another morphism, showcasing the structure and behavior of mappings in algebraic geometry.
5. Morphism provides a way to study how properties of one curve can influence or correspond to properties in another, particularly in families of curves.

Review Questions

• How does a morphism between two curves help us understand their geometric relationships?
  • A morphism between two curves serves as a bridge that connects their respective geometric structures. By mapping points from one curve to another while preserving their algebraic relations, we can analyze how the curves interact with each other. This understanding can reveal properties such as how the curves intersect, their respective genera, and how one curve may influence the characteristics of another.
• Discuss how rational maps differ from regular morphisms and provide an example to illustrate this difference.
  • Rational maps differ from regular morphisms in that they may not be well-defined everywhere on their domain. For example, consider a rational map that is defined by the ratio of two polynomials; it might fail at points where the denominator equals zero. In contrast, a morphism must be well-defined across its entire domain. This distinction highlights the subtleties in working with maps between curves and understanding their behaviors in various contexts.
• Evaluate the significance of dominant morphisms in the context of studying families of curves.
  • Dominant morphisms are significant because they ensure that the image of one curve densely covers part of another curve. This property allows for robust exploration of families of curves since it enables mathematicians to draw conclusions about various members based on structural similarities. For instance, if we have a family of elliptic curves parameterized by a dominant morphism, understanding this connection can lead to insights about moduli spaces and deformation theory, revealing deeper aspects of algebraic geometry.

Related terms

Rational map: A rational map is a function defined on the open subsets of two varieties that may not be well-defined everywhere, often expressed as a quotient of polynomials.

Isomorphism:

An isomorphism between curves is a morphism that is both injective and surjective, establishing a one-to-one correspondence between the points of the two curves.

Image of a morphism: The image of a morphism is the set of points in the target curve that are reached by mapping points from the source curve via the morphism.