Varieties

Definition

In mathematics, particularly in algebraic geometry, varieties are the fundamental objects studied, representing the solution sets of systems of polynomial equations. They can be defined over various fields and can possess different structures, such as affine or projective varieties, which influence their geometric properties and behavior under various transformations.

5 Must Know Facts For Your Next Test

1. Varieties can be classified into affine varieties, which are subsets of affine space defined by polynomials, and projective varieties, which are defined in projective space.
2. The concept of varieties extends beyond just real or complex numbers; they can be defined over any field, including finite fields.
3. The Zariski topology is a key aspect of varieties, where closed sets correspond to algebraic sets defined by polynomial equations.
4. Smooth varieties have well-defined tangent spaces at every point, while singular varieties can have points where these tangent spaces fail to be well-defined.
5. Periodic points in the context of varieties refer to points that return to their original position after applying a certain function repeatedly, highlighting important dynamical behaviors in algebraic geometry.

Review Questions

• How do affine and projective varieties differ in terms of their definitions and properties?
  • Affine varieties are defined as the zero sets of polynomial functions in affine space, while projective varieties arise in projective space where points are represented by homogeneous coordinates. This distinction leads to different geometric properties; for example, projective varieties allow for a more natural treatment of points at infinity, making them essential for studying global properties. Additionally, projective varieties often have richer structure due to the way they handle intersections and dimension.
• Discuss how morphisms between varieties influence their geometric and algebraic properties.
  • Morphisms act as bridges between varieties, allowing us to translate problems from one variety to another while preserving algebraic relationships. They can be thought of as continuous maps that respect the structure of the varieties involved. This means that properties such as dimension and singularities can often be analyzed through their morphisms. Understanding these maps can also lead to important results about the classifications and characteristics of different types of varieties.
• Evaluate the role of periodic points within varieties and how they contribute to understanding dynamical systems in algebraic geometry.
  • Periodic points play a significant role in studying dynamical systems on varieties by providing insights into the behavior of maps that are applied repeatedly. These points can help identify stability or chaos in a system, revealing patterns and structures that may not be evident from a single iteration. By analyzing periodic points within the context of varieties, mathematicians can uncover deeper connections between algebraic geometry and dynamical systems, leading to advancements in both fields.

Related terms

Algebraic Set: A subset of an affine space that is defined as the common solutions to a set of polynomial equations.

Morphisms:

Functions between varieties that preserve the algebraic structure, enabling the study of relationships between different varieties.

Dimension:

A measure of the size or complexity of a variety, indicating how many parameters are needed to describe it locally.