Néron–Séveri group

Definition

The Néron–Séveri group is an important algebraic structure associated with a smooth projective variety over a field, capturing the essential features of its divisor class group. It provides a framework to understand the relations between algebraic cycles and the behavior of divisors, particularly when considering changes in the base field and degeneration phenomena. This group plays a crucial role in studying Néron models, which help extend the properties of varieties from characteristic zero to positive characteristic contexts.

5 Must Know Facts For Your Next Test

1. The Néron–Séveri group is denoted as NS(X) for a smooth projective variety X and is defined as the quotient of the divisor class group by numerical equivalence.
2. The rank of the Néron–Séveri group gives valuable information about the Picard number, which counts the independent classes of line bundles on the variety.
3. It is particularly significant in understanding degeneration phenomena, as it allows one to analyze how divisors behave under changes in families of varieties.
4. In characteristic zero, the Néron–Séveri group can be closely related to the Mordell-Weil group of rational points on abelian varieties.
5. The study of Néron–Séveri groups also plays a key role in formulating conjectures related to variations of Hodge structure in algebraic geometry.

Review Questions

• How does the Néron–Séveri group relate to the divisor class group and why is this relationship important?
  • The Néron–Séveri group is essentially a refinement of the divisor class group, as it takes into account numerical equivalence among divisors. By focusing on this equivalence, we can better understand how divisors behave in families and under various algebraic operations. This relationship is important because it helps classify line bundles and reveals deeper properties of the underlying geometry, especially when extending results from characteristic zero to positive characteristic.
• Discuss how Néron models facilitate the study of Néron–Séveri groups across different characteristics.
  • Néron models provide a way to extend algebraic varieties from characteristic zero to local fields of positive characteristic while preserving their essential geometric properties. By establishing such models, one can study how divisors and their classes behave under degeneration and specialization. The Néron–Séveri group becomes an invaluable tool in this context, allowing mathematicians to analyze connections between the geometry of varieties and their behavior in various characteristics, making it easier to formulate conjectures regarding their structures.
• Evaluate how understanding the Néron–Séveri group contributes to broader theories within arithmetic geometry.
  • Understanding the Néron–Séveri group significantly contributes to various theories within arithmetic geometry by linking divisorial properties with rational points on varieties. It serves as a bridge between algebraic cycles and arithmetic aspects of varieties, especially in contexts involving abelian varieties and their associated Mordell-Weil groups. Furthermore, insights gained from studying these groups can inform larger conjectures related to motives and Hodge theory, thereby connecting seemingly disparate areas within mathematics through this central concept.

Related terms

Divisor Class Group: The group formed by the equivalence classes of divisors on a variety, which captures the notion of linear equivalence among divisors.

Néron Model: A special kind of model of an algebraic variety defined over a non-Archimedean local field, which allows for a good extension of the variety to the integral model.

Algebraic Cycle: A formal sum of subvarieties of a given variety, with coefficients in integers, used to study the intersection theory and the Chow ring.