Discrepancies

Definition

Discrepancies refer to the differences or inconsistencies that arise between two mathematical objects, often indicating an error or a need for adjustment in the context of algebraic geometry. They play a significant role when examining singularities and are crucial in the process of blowing up to achieve resolution. Understanding discrepancies helps in analyzing how singularities affect the properties of algebraic varieties and contributes to resolving these singularities effectively.

5 Must Know Facts For Your Next Test

1. Discrepancies are calculated using the formula involving the discrepancy divisor, which measures how far a given divisor deviates from being effective in the resolution process.
2. They are closely linked to the concept of minimal models, where discrepancies help determine whether a variety can be simplified further.
3. In resolution techniques, discrepancies can indicate the number of times a blow-up is needed at a particular point to achieve regularity.
4. The discrepancies can be positive, zero, or negative, which gives information about the nature and type of singularity present in the variety.
5. Understanding discrepancies is fundamental to the theory of minimal models and plays a key role in classifications within algebraic geometry.

Review Questions

• How do discrepancies relate to the process of blowing up and what role do they play in resolving singularities?
  • Discrepancies are essential when blowing up because they indicate how much adjustment is needed at specific points to correct singularities. When a blow-up occurs, discrepancies help measure the difference between the original and transformed varieties. They provide vital information on how many times and where blow-ups must happen to ensure that the resulting variety becomes non-singular, ultimately leading to effective resolution.
• Discuss how discrepancies impact the classification of algebraic varieties during the resolution of singularities.
  • Discrepancies significantly influence the classification of algebraic varieties because they determine whether a variety can be simplified into a minimal model. By assessing discrepancies, mathematicians can identify which varieties are effectively resolvable and categorize them based on their singularity types. This classification informs further research and applications in algebraic geometry by revealing structural properties of varieties that are crucial for understanding their behavior.
• Evaluate the implications of discrepancies on the study of minimal models and their importance in algebraic geometry.
  • Discrepancies have profound implications for the study of minimal models since they provide insights into how far a given variety is from achieving minimality. By analyzing discrepancies, one can determine whether further blow-ups are necessary or if a variety is already in its simplest form. This evaluation aids in understanding complex relationships between varieties and their resolutions, making discrepancies pivotal for advancing research in algebraic geometry and its applications.

Related terms

Blowing Up: A transformation that replaces a point or a subvariety in an algebraic variety with a projective space, helping to resolve singularities.

Singularity: A point at which a mathematical object is not well-behaved, typically where functions fail to be differentiable or where varieties do not meet certain regularity conditions.

Resolution of Singularities:

A process that transforms a singular variety into a non-singular one through a series of blow-ups, addressing discrepancies along the way.