Discrepancies

Definition

In algebraic geometry, discrepancies are numerical invariants associated with a variety that help classify its singularities, specifically relating to how the singularity differs from a smooth variety. They play a crucial role in the Minimal Model Program (MMP) and are essential for understanding the nature of singularities, particularly when examining canonical and terminal singularities. The notion of discrepancies serves as a bridge to various resolutions and provides insight into the geometric structure of varieties.

5 Must Know Facts For Your Next Test

1. Discrepancies are defined for a log pair $(X, D)$ where $X$ is a variety and $D$ is a divisor; they quantify how much the pair deviates from being canonical.
2. A variety with non-negative discrepancies is called canonical, while those with positive discrepancies are termed terminal, indicating different geometric properties.
3. Discrepancies can be computed using the formula involving the intersection numbers of the divisor with the resolution of singularities.
4. The discrepancy can change under birational transformations, making it an important tool in understanding how varieties behave under different resolutions.
5. They are used to determine whether certain transformations will result in a minimal model, thus playing a vital role in the Minimal Model Program.

Review Questions

• How do discrepancies relate to the classification of singularities in algebraic geometry?
  • Discrepancies provide a numerical measure that helps classify singularities by indicating how far a variety deviates from being smooth or canonical. For instance, when examining a log pair $(X, D)$, if the discrepancies are negative, it suggests that the variety has worse singularities than those classified as canonical. This classification helps determine whether one is dealing with terminal or canonical singularities, which are crucial for understanding the geometry of varieties.
• Discuss the significance of discrepancies in the context of resolutions of singularities.
  • Discrepancies play a critical role in resolutions of singularities as they help assess how singular points transform under birational maps. During resolution processes, discrepancies can change based on the types of modifications made to the variety. Understanding these changes is key because they indicate whether further adjustments are needed to achieve a smooth model or if the singularity is adequately resolved.
• Evaluate the impact of discrepancies on the Minimal Model Program and how they influence the pursuit of minimal models.
  • Discrepancies significantly influence the Minimal Model Program (MMP) by providing essential criteria for determining whether a variety can be transformed into a minimal model. When analyzing varieties through this program, discrepancies help identify regions where the geometry may need refining or altering. If a variety has negative discrepancies, it suggests that it cannot yet be considered minimal and must undergo further resolution or transformation until all discrepancies are non-negative, thus guiding the path toward achieving minimality.

Related terms

Singularity: A point at which a mathematical object is not well-behaved, often characterized by properties like non-removability or non-uniqueness in its structure.

Kawamata-Viehweg Vanishing Theorem: A result in algebraic geometry that provides conditions under which certain cohomology groups vanish, often utilized in the study of discrepancies and resolutions.

Resolution of Singularities: A process by which a singular variety is transformed into a non-singular variety, allowing for the study and understanding of its properties and discrepancies.