Projective variety

Definition

A projective variety is a type of geometric object that is defined as the zero set of homogeneous polynomials in projective space. These varieties can be thought of as the set of points that satisfy certain polynomial equations, and they play a central role in algebraic geometry, especially when exploring the properties and relationships of algebraic objects through geometric means.

5 Must Know Facts For Your Next Test

1. Projective varieties can be thought of as generalizations of affine varieties, extending the notion of solutions to polynomial equations to include points at infinity.
2. Every projective variety is defined within a projective space, which is crucial for understanding its geometric and algebraic properties.
3. The dimension of a projective variety is determined by the number of homogeneous coordinates needed to describe it.
4. Projective varieties are essential in studying vector bundles, particularly in applications such as the Borel-Weil theorem where they provide a geometric realization of line bundles.
5. The intersection theory in projective varieties allows for the computation of classes in cohomology, linking algebraic geometry with topological concepts.

Review Questions

• How do projective varieties extend the concept of affine varieties, and why is this extension significant in algebraic geometry?
  • Projective varieties extend affine varieties by incorporating points at infinity, allowing them to capture the complete geometric picture defined by polynomial equations. This extension is significant because it provides a more comprehensive framework to study solutions to polynomial equations, helping to analyze their properties under transformations. By studying projective varieties, we can understand various aspects of algebraic geometry that are not evident when limiting ourselves to affine settings.
• Discuss the role of projective varieties in the Borel-Weil theorem and how they facilitate geometric realizations of line bundles.
  • Projective varieties serve as the underlying spaces in which the Borel-Weil theorem operates, providing a geometric context for line bundles associated with representations of algebraic groups. This theorem demonstrates how line bundles can be realized geometrically as sections over projective varieties, leading to connections between algebraic and geometric properties. Such realizations help bridge the gap between abstract algebraic concepts and their visual interpretations in geometry.
• Evaluate the impact of cohomology on understanding projective varieties, particularly in terms of intersection theory and its applications.
  • Cohomology significantly enhances our understanding of projective varieties by providing tools to analyze their topological and algebraic properties through intersection theory. By examining how projective varieties intersect, we can derive valuable information about their classes in cohomology, revealing deeper structural insights. The interplay between cohomology and projective varieties not only enriches algebraic geometry but also establishes connections with other areas like topology and representation theory, showcasing its broad relevance.

Related terms

Homogeneous polynomial: A polynomial in which all terms have the same total degree, crucial for defining projective varieties since they are formed as the vanishing sets of such polynomials.

Projective space: A space obtained by taking a vector space and adding 'points at infinity' to it, allowing us to study properties that remain invariant under scaling of coordinates.

Cohomology:

A mathematical tool used to study the topological properties of varieties, providing insights into their structure and relationships.