Terminal Singularity

Definition

A terminal singularity is a type of singularity in algebraic geometry where the local ring has a particular structure that implies it cannot be 'resolved' in a certain sense. This means that the singular point cannot be transformed into a smoother point through a sequence of blow-ups without introducing new singularities. Terminal singularities are important for understanding the minimal model program, as they represent one of the conditions that help categorize varieties based on their singular structures.

5 Must Know Facts For Your Next Test

1. Terminal singularities can occur in varieties of any dimension and play a key role in the classification of algebraic varieties.
2. These singularities are considered 'mild' compared to other types, meaning they have certain desirable properties for resolution processes.
3. In the context of the minimal model program, terminal singularities are often used to identify which varieties can be transformed into minimal models.
4. The existence of a terminal singularity indicates that the variety is not Fano but rather belongs to a different category of varieties with specific geometric properties.
5. Terminal singularities can be characterized by specific inequalities involving their discrepancies, which provide important information about their structure.

Review Questions

• How do terminal singularities relate to the resolution of singularities in algebraic geometry?
  • Terminal singularities are significant because they represent cases where resolution is particularly well-behaved. In the resolution process, these types of singularities do not introduce new complications during blow-ups, making them easier to handle. Understanding terminal singularities helps mathematicians determine when they can successfully resolve a variety into a smoother form without encountering additional issues.
• Discuss the implications of terminal singularities on the minimal model program and the classification of algebraic varieties.
  • Terminal singularities are critical in the minimal model program as they help categorize varieties based on their geometric and algebraic properties. Varieties with terminal singularities are often considered for transformation into minimal models since these singular points indicate a level of 'mildness' that makes them suitable for simplification. The presence of such singularities allows researchers to classify varieties more effectively, distinguishing between those that can achieve minimal status versus those that cannot.
• Evaluate the role of terminal singularities in understanding the discrepancies associated with algebraic varieties and their resolutions.
  • Terminal singularities play an essential role in evaluating discrepancies within algebraic varieties as they provide insight into how these discrepancies behave under various transformations. By studying terminal singularities, mathematicians can derive inequalities that reveal crucial information about the structure and classification of varieties. This understanding not only aids in determining whether a variety can be resolved but also helps establish broader connections between geometric properties and birational equivalence in algebraic geometry.

Related terms

Resolution of Singularities: The process of transforming a singular variety into a nonsingular one through blow-ups and other modifications.

Minimal Model Program: A program in algebraic geometry aimed at classifying algebraic varieties by examining their birational properties and resolving singularities.

Kawamata-Viehweg Vanishing Theorem: A fundamental result in algebraic geometry that provides conditions under which certain cohomology groups vanish, influencing the study of singularities.