Base locus

Definition

The base locus is the set of points in a projective space where a linear system fails to be effective, meaning that it represents a certain failure in the ability to generate sections of a line bundle. This set plays a crucial role in understanding the properties of linear systems and how they relate to the geometry of varieties, influencing both their dimension and the behavior of their morphisms.

5 Must Know Facts For Your Next Test

1. The base locus is defined as the intersection of the supports of all effective divisors associated with a linear system, providing insight into where sections fail to exist.
2. If the base locus is empty, it indicates that the linear system is complete and provides many sections that can be used effectively throughout the entire projective space.
3. For a given linear system, knowing its base locus helps determine properties like dimensionality and can reveal hidden symmetries within the variety.
4. The base locus can vary under the action of automorphisms or deformations of the variety, which can lead to changes in how linear systems are realized.
5. In the study of morphisms, understanding the base locus can give rise to effective results on how maps from one variety to another behave, particularly regarding their injectivity or other critical features.

Review Questions

• How does the base locus affect the effectiveness of a linear system on a projective space?
  • The base locus directly indicates where a linear system fails to provide effective sections. When points lie within the base locus, it means those points are not represented by any section in the linear system. Thus, understanding the base locus allows us to assess which parts of the projective space remain inaccessible or insufficiently covered by sections generated from the linear system.
• Compare and contrast a linear system with an empty base locus and one with non-empty base locus in terms of their geometric implications.
  • A linear system with an empty base locus suggests that every point in projective space is effectively covered by sections of that system, leading to rich geometric structures and often indicating ample global sections. In contrast, a non-empty base locus implies certain restrictions and limitations on the coverage of those points, potentially leading to singularities or other complexities in the geometry. This difference highlights how crucial it is to analyze the base locus when studying varieties and their morphisms.
• Evaluate the role of the base locus in determining morphisms between varieties, including its implications for injectivity and mapping behavior.
  • The base locus plays a significant role in determining morphisms by revealing areas where mappings may collapse or lose information. If a morphism from one variety to another has its base locus captured within it, this can suggest that certain fibers might be non-injective or have more complicated structures than anticipated. Thus, assessing how the base locus behaves under various mappings can provide deeper insights into potential singularities or obstructions in achieving a clean morphism between varieties.

Related terms

linear system: A linear system is a family of effective divisors on a variety that can be expressed as the complete linear combination of a given divisor, capturing the way sections can be combined.

effective divisor: An effective divisor is a formal sum of subvarieties with non-negative integer coefficients, representing sections of a line bundle that have associated meromorphic functions.

line bundle: A line bundle is a vector bundle of rank one over a variety, which allows for the study of sections and divisors within the context of algebraic geometry.