Canonical Divisor

Definition

A canonical divisor is a divisor on an algebraic variety that captures the geometric and arithmetic properties of the variety, particularly its singularities and its behavior under various morphisms. It is an important concept in the context of the adjunction formula, which relates the canonical divisor of a variety to its subvarieties and their respective canonical divisors. Canonical divisors play a crucial role in the study of the resolution of singularities and in understanding how varieties can be transformed to achieve better geometric properties.

5 Must Know Facts For Your Next Test

1. The canonical divisor is often denoted by $K_X$ for a variety $X$, representing its class in the divisor group.
2. In a smooth projective variety, the canonical divisor is effective if the variety has non-negative Kodaira dimension.
3. The behavior of the canonical divisor under blow-ups can be crucial in resolving singularities, as it transforms into a different divisor after the blow-up process.
4. The calculation of canonical divisors is essential in understanding various invariants of varieties, such as their Kodaira dimension and plurigenera.
5. Canonical divisors help classify varieties into different types based on their singularities, such as rational or log terminal singularities.

Review Questions

• How does the concept of a canonical divisor relate to the resolution of singularities in algebraic geometry?
  • The canonical divisor is closely tied to the resolution of singularities because it provides essential information about how singularities behave under morphisms. When resolving singularities through processes like blowing up, the canonical divisor often changes, reflecting the new structure of the variety. Understanding this transformation helps mathematicians analyze how singular points can be smoothed out and how this impacts overall geometry.
• Discuss the importance of the adjunction formula in calculating canonical divisors for subvarieties.
  • The adjunction formula is vital for calculating canonical divisors because it establishes a relationship between the canonical divisor of a variety and those of its subvarieties. This formula allows us to express the canonical divisor of a subvariety in terms of the ambient variety's canonical divisor, aiding in computations. It’s particularly useful when analyzing complex structures and understanding how subvarieties contribute to the overall geometric properties of a larger variety.
• Evaluate how variations in the canonical divisor impact the classification of algebraic varieties with respect to their singularities.
  • Variations in the canonical divisor directly influence how algebraic varieties are classified regarding their singularities. For instance, if a canonical divisor is effective, it indicates that certain conditions are satisfied, such as having non-negative Kodaira dimension. This classification helps mathematicians determine whether a variety is rational or log terminal based on its singularity type. The changes in the canonical divisor during processes like resolution provide insights into which classes of varieties can be smoothed or transformed, shaping our understanding of their geometric properties.

Related terms

Divisor: A formal sum of subvarieties of codimension one, used to generalize the notion of functions having zeros and poles on a variety.

Adjunction Formula: A key result in algebraic geometry that relates the canonical divisor of a variety to those of its subvarieties and helps compute them under certain conditions.

Resolution of Singularities: A process that replaces a singular variety with a smoother one, often involving blowing up points or subvarieties to study their properties better.