Effective Divisor

Definition

An effective divisor is a formal sum of points on an algebraic variety with non-negative integer coefficients, which essentially represents a way to count the number of times each point appears. This concept is key in understanding how divisors can be used to describe linear systems and their properties, especially when dealing with Riemann-Roch theorem applications and minimal models in birational geometry.

5 Must Know Facts For Your Next Test

1. An effective divisor can be expressed as $$D = ext{sum}(n_i P_i)$$, where each $$n_i$$ is a non-negative integer and $$P_i$$ are points on the variety.
2. In the context of linear systems, effective divisors play a crucial role in determining the dimension of the space of global sections of line bundles.
3. The Riemann-Roch theorem connects effective divisors with cohomology groups, providing important insights into the dimensions of these spaces.
4. Effective divisors can be seen as a way to keep track of the zeros and poles of meromorphic functions on the variety.
5. In birational geometry, effective divisors are fundamental in studying minimal models, as they help classify and compare different geometrical structures on varieties.

Review Questions

• How do effective divisors relate to Weil and Cartier divisors in algebraic geometry?
  • Effective divisors are specific instances of Weil divisors, which can include negative coefficients. While Weil divisors allow for more general combinations, effective divisors focus on non-negative counts. Cartier divisors provide an algebraic perspective that can also relate back to effective divisors since they can be defined using sections that involve these non-negative coefficients. The relationship highlights how different types of divisors help us understand the geometry of varieties.
• What role do effective divisors play in determining the dimensions of linear systems on an algebraic variety?
  • Effective divisors are essential in defining linear systems because they determine which global sections of line bundles exist. The dimension of a linear system corresponds to the number of effective divisors that are linearly equivalent to each other. This allows us to analyze how many independent meromorphic functions can exist on the variety based on these effective counts. Understanding this connection helps reveal the richness of the variety's geometry.
• Evaluate the impact of effective divisors on the application of the Riemann-Roch theorem for curves and surfaces.
  • Effective divisors are pivotal in applying the Riemann-Roch theorem because they directly affect the calculation of dimensions related to cohomology groups. The theorem gives a formula that relates the dimension of global sections to various characteristics associated with effective divisors, including their degree and genus. This connection is crucial in understanding how these dimensions vary across different types of curves and surfaces, illustrating the interplay between algebraic properties and geometric structures.

Related terms

Weil Divisor: A Weil divisor is a formal sum of irreducible subvarieties of codimension one, allowing for both positive and negative integer coefficients, used to generalize the concept of divisors in algebraic geometry.

Cartier Divisor:

A Cartier divisor is a locally defined divisor that can be represented by a global section of a line bundle, which provides a more algebraic approach to understanding divisors.

Linear System: A linear system is the set of effective divisors that are linearly equivalent to a given divisor, reflecting the space of rational functions associated with it.