Effective Divisors And Triviality Exploring Linear And Numerical Equivalence

In the realm of algebraic geometry, divisors play a pivotal role in understanding the geometry of algebraic varieties. This article delves into a fundamental question regarding the nature of effective divisors: Can they be linearly or numerically trivial? This exploration is crucial for grasping the intricate relationships between divisors, linear equivalence, numerical equivalence, and the properties of algebraic varieties, particularly in the context of rational connectedness.

Understanding Divisors and Their Significance

Before we tackle the central question, it's essential to establish a solid understanding of what divisors are and why they matter in algebraic geometry.

Divisors, at their core, are formal sums of codimension-one subvarieties (prime divisors) on an algebraic variety. They provide a powerful tool for studying the geometry of the variety by encoding information about subvarieties and their multiplicities. Divisors are fundamental objects in algebraic geometry, serving as a powerful tool for understanding the geometric properties of algebraic varieties. A divisor on a variety X is essentially a formal linear combination of irreducible subvarieties of codimension one. These subvarieties, often called prime divisors, are the building blocks of the divisor theory. The coefficients in the linear combination indicate the multiplicity of each subvariety. Divisors provide a way to study the intersection theory of varieties and to understand the behavior of rational functions and morphisms. The study of divisors allows us to analyze the behavior of rational functions, morphisms, and intersection theory on the variety. They act as a bridge between the algebraic and geometric aspects of the space, offering a way to translate geometric intuition into algebraic language, and vice versa. The concept of a divisor allows us to generalize the notion of zeros and poles of a function to higher dimensions, providing a sophisticated way to track the behavior of functions on the variety. The group of divisors carries a rich algebraic structure, and its properties reflect the underlying geometry of the variety. For instance, the Picard group, which classifies line bundles up to isomorphism, is closely related to the group of divisors modulo linear equivalence. By studying divisors, we can gain insights into the birational geometry of the variety, including its singularities, its cycles, and its rational maps to other varieties.

Linear Equivalence and Numerical Equivalence

Two critical concepts related to divisors are linear equivalence and numerical equivalence. These equivalence relations help us classify divisors based on their geometric and intersection-theoretic properties.

Linear equivalence, the stronger of the two, relates divisors that differ by the divisor of a rational function. Two divisors, D and E, are said to be linearly equivalent (denoted D ~ E) if their difference (D - E) is the divisor of a rational function f on the variety. In other words, D - E = div(f). This means that the zeros and poles of f precisely account for the difference between the multiplicities of the prime divisors in D and E. Linear equivalence captures the idea that divisors in the same linear equivalence class have similar geometric behavior, as they are essentially the same up to the addition of zeros and poles of a rational function. The study of linear equivalence classes is closely linked to the study of line bundles and their sections, which provide a powerful tool for understanding the geometry of the variety. The set of all divisors linearly equivalent to a given divisor forms a linear system, which can be thought of as a family of divisors parametrized by a projective space. The base locus of a linear system, which is the set of points where all divisors in the system intersect, provides valuable information about the geometry of the variety. Understanding linear equivalence is crucial for many aspects of algebraic geometry, including the classification of varieties, the study of their birational geometry, and the construction of moduli spaces.

Numerical equivalence, on the other hand, is a weaker equivalence relation that focuses on intersection numbers. Two divisors, D and E, are numerically equivalent (denoted D ≡ E) if their intersection numbers with any curve C on the variety are the same: D · C = E · C. In essence, numerical equivalence captures the idea that two divisors have the same intersection behavior with all curves on the variety. Numerical equivalence is a weaker notion than linear equivalence, but it still plays a crucial role in algebraic geometry. Two divisors are numerically equivalent if their intersection numbers with all curves on the variety are the same. This means that, while they may not be related by the divisor of a rational function, they have the same overall intersection behavior. Numerical equivalence classes capture the essential intersection-theoretic properties of divisors, and they are closely related to the cone of effective cycles and the cone of nef cycles. The Néron-Severi group, which is the group of divisors modulo numerical equivalence, is a finitely generated abelian group that reflects the intersection-theoretic structure of the variety. Studying numerical equivalence allows us to understand the ampleness and nefness of divisors, which are crucial concepts in the classification of varieties. A divisor is ample if some multiple of it defines an embedding of the variety into projective space, while a divisor is nef if its intersection number with any curve is non-negative. These notions are fundamental for understanding the positivity properties of divisors and their role in the geometry of the variety. Numerical equivalence also plays a key role in the Minimal Model Program, which is a central tool in the birational classification of algebraic varieties. It emphasizes the intersection-theoretic properties of divisors, making it particularly relevant for studying the global geometry of the variety. While linear equivalence is tied to the local behavior of rational functions, numerical equivalence reflects the overall intersection pattern of divisors with curves on the variety. The relationship between linear and numerical equivalence is a central theme in algebraic geometry, and understanding this relationship is crucial for many applications.

Effective Divisors

An effective divisor is a divisor where all the coefficients in the formal sum are non-negative. Effective divisors are a fundamental concept in algebraic geometry, representing divisors that are non-negative linear combinations of prime divisors. In simpler terms, an effective divisor corresponds to a subvariety of the algebraic variety, possibly with multiplicities. They are crucial for understanding the geometry and topology of the variety and play a vital role in various aspects of algebraic geometry, such as intersection theory, linear systems, and the study of ample and nef divisors. The notion of effectiveness is closely related to the idea of positivity in the context of divisors. An effective divisor can be thought of as a divisor that represents a