Projective Varieties

Definition

Projective varieties are the geometric manifestations of algebraic equations in projective space, extending the notion of affine varieties to include 'points at infinity.' They are defined as the common zeros of a set of homogeneous polynomials in projective coordinates. These varieties allow for a richer geometric structure and enable the exploration of properties that remain invariant under projective transformations, connecting algebraic geometry and geometric concepts in a profound way.

5 Must Know Facts For Your Next Test

1. Projective varieties are defined over projective space $$ ext{P}^n$$, which consists of lines through the origin in $$ ext{K}^{n+1}$$, where $$ ext{K}$$ is a field.
2. Every affine variety can be embedded into a projective variety by adding points at infinity, demonstrating how projective varieties can encapsulate affine ones.
3. The intersection of two projective varieties is itself a projective variety, highlighting their stability under geometric operations.
4. Projective varieties possess unique properties such as being complete, meaning they contain all limit points, unlike affine varieties which can be open.
5. The concept of dimension in projective varieties corresponds to the dimension of the underlying projective space and is determined by the number of independent equations defining them.

Review Questions

• How do projective varieties extend the concept of affine varieties, particularly regarding points at infinity?
  • Projective varieties broaden the idea of affine varieties by incorporating points at infinity, which allows for a more complete understanding of geometric properties. While affine varieties are limited to finite points defined by polynomial equations, projective varieties include all lines through the origin in a higher-dimensional space. This inclusion is essential for studying certain geometric behaviors and transformations that cannot be captured by affine varieties alone.
• Discuss the importance of homogeneous polynomials in defining projective varieties and how they relate to the structure of projective space.
  • Homogeneous polynomials play a critical role in defining projective varieties because they ensure that the equations describing these varieties maintain their form under scalar multiplication. In projective space, each point is represented by homogeneous coordinates, making it essential that the polynomials used are homogeneous. This relationship allows for consistency across different representations and helps define the geometric structure that characterizes projective varieties.
• Evaluate how the properties of completeness and intersection behavior in projective varieties impact their study compared to affine varieties.
  • The completeness property of projective varieties indicates that they include all limit points, which is crucial for various aspects of algebraic geometry such as compactness and cohomology. This stands in contrast to affine varieties, which may have 'holes' or lack limit points. Furthermore, since intersections of projective varieties yield other projective varieties, this stability supports deeper investigations into their structure and relationships. Such properties facilitate more sophisticated analyses and applications within algebraic geometry compared to their affine counterparts.

Related terms

Homogeneous Polynomials: Polynomials whose terms all have the same total degree, which are crucial for defining projective varieties.

Projective Space:

A space that extends the concept of Euclidean space by adding 'points at infinity,' providing a framework for studying projective varieties.

Affine Varieties:

Varieties defined by polynomial equations in affine space, serving as a foundational concept before extending to projective varieties.