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 solutions to equations that describe geometric shapes, and they are important because they allow us to work with properties that are invariant under projective transformations, making them a central concept in algebraic geometry.

5 Must Know Facts For Your Next Test

1. Projective varieties are defined as the zero sets of homogeneous polynomials, which means all terms in the defining equations have the same degree.
2. These varieties can be viewed as subsets of projective space, and they allow for a more elegant treatment of intersections and other properties compared to affine varieties.
3. One key feature of projective varieties is that they are compact, meaning they can be covered by a finite number of open sets, which simplifies many geometric considerations.
4. The study of projective varieties leads to the Picard group, which classifies line bundles over these varieties, emphasizing their role in understanding divisors and their associated sheaves.
5. In geometric invariant theory, projective varieties serve as the foundation for studying objects under group actions, allowing us to classify orbits and understand moduli spaces.

Review Questions

• How does the definition of a projective variety relate to homogeneous polynomials and projective space?
  • A projective variety is fundamentally defined by the zero set of homogeneous polynomials within projective space. This means that to describe these varieties, we consider polynomials where each term has the same degree. The notion of projective space introduces points at infinity, allowing us to interpret the solutions geometrically without losing information about intersections or limits, which is essential for understanding their structure and properties.
• Discuss the significance of compactness in projective varieties and how it contrasts with affine varieties.
  • Compactness in projective varieties is significant because it means these varieties can be covered by finitely many open sets, which is not necessarily true for affine varieties. This property allows for easier application of various geometric and topological tools. While affine varieties can extend infinitely, projective varieties encapsulate all their limit points within a finite framework, leading to richer mathematical structures and better control over their geometric behavior.
• Evaluate the role of projective varieties in connecting algebraic geometry with geometric invariant theory.
  • Projective varieties serve as a crucial link between algebraic geometry and geometric invariant theory by providing a structured environment where we can study algebraic objects under group actions. These varieties allow us to classify orbits formed by these actions and explore moduli spaces, which represent families of similar algebraic objects. Through this connection, we can gain insights into both the geometric aspects of these objects and their algebraic properties, showcasing the deep interplay between geometry and algebra.

Related terms

Homogeneous polynomial: A polynomial whose terms all have the same total degree, which is crucial for defining projective varieties since they are described by such polynomials.

Projective space: A space that formalizes the idea of points at infinity, where lines through the origin in a vector space correspond to points in projective space, allowing us to study geometrical properties without worrying about affine limitations.

Closure in Zariski topology: The closure of a set of points in algebraic geometry which helps to define varieties by taking into account their limit points and establishing the relationship between projective and affine varieties.