Exceptional divisor

Definition

An exceptional divisor is a type of divisor that arises during the process of blowing up a variety at a point, which resolves singularities. In essence, it represents the preimage of the point where the blow-up occurs and provides a way to understand the structure of the modified variety. This concept is crucial for analyzing how singularities can be 'resolved' or altered into a smoother geometric form.

5 Must Know Facts For Your Next Test

1. Exceptional divisors are generally smooth and play an important role in understanding the birational geometry of varieties.
2. In the context of blowing up at a point, an exceptional divisor corresponds to the set of points that lie over the original singular point being blown up.
3. Every blow-up introduces one exceptional divisor for each point that is blown up, leading to changes in the geometric properties of the space.
4. Exceptional divisors have self-intersection numbers that can often be used to classify and study singularities in algebraic varieties.
5. The concept of exceptional divisors is fundamental in higher-dimensional algebraic geometry, particularly in studying minimal models and Mori theory.

Review Questions

• How does the exceptional divisor relate to the process of blowing up a variety and what role does it play in resolving singularities?
  • When blowing up a variety at a point, the exceptional divisor captures the information about how that singularity is transformed into a more manageable structure. Specifically, it replaces the blown-up point with a projective space, allowing us to study the local behavior around that point more effectively. The exceptional divisor helps us visualize and analyze how singularities can be resolved, providing insights into the overall geometry of the modified variety.
• Discuss the significance of self-intersection numbers of exceptional divisors in classifying singularities in algebraic geometry.
  • Self-intersection numbers of exceptional divisors are critical in understanding the nature and complexity of singularities. These numbers provide quantitative data about how divisors interact within the geometry of varieties, and they can indicate whether singularities are mild or severe. By studying these self-intersections, mathematicians can classify different types of singularities and determine appropriate resolutions, which is key for effective analysis in algebraic geometry.
• Evaluate the implications of introducing multiple exceptional divisors when performing successive blow-ups on a variety.
  • When multiple exceptional divisors are introduced through successive blow-ups, it significantly alters the geometric structure and birational properties of the variety. Each new blow-up adds complexity and can lead to various interactions between divisors, impacting resolution strategies for singularities. Analyzing these implications allows mathematicians to understand better how different resolutions affect not just individual points but also broader geometric features, providing valuable insight into both local and global properties of varieties.

Related terms

Blow-up: A transformation of a variety that replaces a point with an entire projective space, allowing for the resolution of singularities.

Resolution of singularities: A process by which singular points of a variety are replaced with smoother points to make the variety easier to study.

Divisor: A formal sum of codimension 1 subvarieties, which can be thought of as a way to keep track of how certain points contribute to the geometry of a space.