Canonical singularities

Definition

Canonical singularities are a type of singularity in algebraic geometry that behaves well under certain mathematical operations, particularly in the context of resolution of singularities. They are significant because they allow for the classification of varieties based on their singular points and play a crucial role in the study of minimal models and the Minimal Model Program. Canonical singularities appear when the canonical divisor has non-negative discrepancies, meaning that these singularities are considered to have mild behavior compared to other types of singularities.

5 Must Know Facts For Your Next Test

1. Canonical singularities are a central concept in algebraic geometry due to their role in understanding the structure of algebraic varieties.
2. They are defined by the property that the discrepancies of the canonical divisor are non-negative, which indicates their mild nature.
3. One important aspect of studying canonical singularities is their relationship with minimal models; varieties with canonical singularities can often be transformed into minimal models.
4. Canonical singularities are significant in the context of birational geometry and play an essential role in proving many important results in this field.
5. The existence of canonical singularities influences various properties of algebraic varieties, including their dimension, connectedness, and other topological features.

Review Questions

• How do canonical singularities differ from terminal singularities, and what implications does this have for their classification?
  • Canonical singularities differ from terminal singularities primarily in terms of discrepancies; canonical singularities have non-negative discrepancies, while terminal singularities have positive discrepancies. This distinction is crucial because it affects how these singularities can be resolved and classified within algebraic geometry. As such, understanding this difference helps mathematicians apply various techniques and tools when working with different types of singular points on varieties.
• Discuss the importance of discrepancies in the study of canonical singularities and their impact on resolutions of singularities.
  • Discrepancies play a vital role in studying canonical singularities as they provide a measure of how 'bad' or 'mild' a singularity is. In particular, non-negative discrepancies indicate that we can resolve these singularities using certain geometric techniques without drastically changing the structure of the variety. This is important for advancing theories like the Minimal Model Program, where resolving canonical singularities allows for simplification and better understanding of algebraic varieties.
• Evaluate the significance of canonical singularities in birational geometry and their broader implications in algebraic geometry.
  • Canonical singularities hold considerable significance in birational geometry as they enable mathematicians to classify varieties based on their geometric properties and understand their birational equivalence. The ability to handle these mild singularities effectively leads to important results, such as the existence of minimal models and impacts on more complex problems in algebraic geometry. Moreover, studying these singular points helps build a deeper understanding of the structure and behavior of higher-dimensional varieties, influencing research across various areas within mathematics.

Related terms

Terminal Singularities: These are a subclass of singularities that are even milder than canonical singularities, characterized by positive discrepancies in their canonical divisor.

Discrepancy: This term refers to a measure of how much the canonical divisor differs from a certain expected divisor when resolving singularities.

Minimal Model Program: A strategy in algebraic geometry aimed at transforming varieties into simpler forms while preserving their essential features, often using resolutions of singularities.