PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Geometry → Algebraic Geometry ↓


Divisors and Intersections


Algebraic geometry is a rich field that combines algebra, geometry, and topology. In this topic, we explore the concepts of divisors and intersections, which play a key role in understanding the structure of algebraic varieties. The aim of this article is to present these ideas in a simple way, so that they are accessible even to those new to the subject.

Separators: The basics

A divisor is a formal sum of subvarieties of co-dimension one in a given variety. Suppose we have an algebraic variety ( V ). A divisor on ( V ) is often written as:

D = a_1 V_1 + a_2 V_2 + ldots + a_n V_n

Here, ( V_1, V_2, ldots, V_n ) are invariant subvarieties of co-dimension one, and ( a_1, a_2, ldots, a_n ) are integers. This idea is similar to linear combinations in algebra, where each ( V_i ) is weighted by an integer coefficient ( a_i ).

Example of separators

Suppose we have two curves in the plane, where one is a line denoted by ( L ) and the other is a circle denoted by ( C ). The separator can be constructed as follows:

D = 2L - 3C

This means that our separator is made up of twice the line and three times the circle.

Linear equivalence of divisors

Two divisors ( D ) and ( D' ) on a variety ( V ) are called linearly equivalent if there exists a rational function ( f ) on ( V ) such that:

D' = D + text{div}(f)

where (text{div}(f)) denotes the divisors of the zeros and poles of the function ( f ). Linear equivalence indicates that the "shapes" or algebraic representations defined by these divisors have the same degrees of freedom with respect to functions on a variety.

Visual example: Linear equivalence

Imagine two different paths on a surface that can be continuously deformed into one another, without crossing singularities or boundaries. The concept of linear equivalence is similar in that two separators can be transformed into one another by adjusting a rational function.

Divisor D = 2L Denominator C = -3C

In the above illustration, the blue line represents the divisor L, and the green circle represents the divisor C. With linear equivalence, both forms can be transferred into each other through a change in rational functions.

Intersection theory

Intersection theory involves studying the ways in which different subvarieties intersect within an algebraic variety. Understanding these intersections helps us understand the structure and behavior of the variety.

Intersection number

The intersection number of two divisors (D) and (D') on a variety is a numerical measure of how these divisors intersect within that variety. This is an important concept because it gives insight into the geometry of varieties.

For example, the intersection number can tell us the number of points where two curves meet within a surface. Mathematically, for divisors ( D ) and ( D' ), the intersection number is represented as ( D cdot D' ).

Example of intersectionality theory

Consider an algebraic surface consisting of two curves ( C_1 ) and ( C_2 ). Their intersection in terms of their explicit expressions might look like this:

C_1: y = x^2 + 3x + 2
C_2: y = -2x^2 + 5

The intersection points of these curves can be found by solving the equations simultaneously, which will indicate where these curves intersect in the plane.

Visualization of intersections

Looking at intersections gives a better intuition for the concept. Here is a representation where two curves intersect at two distinct points on the plane:

In this illustration, the red dashed line and the blue quadratic curve intersect at two points, marked by black dots. These points represent the solutions to the system of equations.

Separator class

In more advanced studies, divisors are considered not individually but as part of an entire equivalence class. A divisor class is a set of divisors that are linearly equivalent to each other. This concept helps to simplify and unify calculations in complex varieties.

Furthermore, rather than focusing on specific separators, researchers often work with these classes, in order to exploit the properties of the varieties without getting entangled in minor differences.

Intersection product

The next layer of complexity involves intersection products, which extend the concept of intersection numbers. In intersection theory, intersection products are a way of defining the "product" of two or more divisors by considering their intersection.

The intersection product is a bilinear form on the space of divisors, and it provides rich algebraic and geometric information. It takes two divisors, often represented as line bundles, and constructs another class.

Mathematically, if we have two line bundles ( L ) and ( M ), then their intersection can be represented as:

[L] cap [M]

Technological and computational impact

Intersection products are particularly important in computational algebraic geometry, where software packages and algorithms are developed to compute and explore these intersections. Understanding them is important in fields as diverse as number theory, topology, and even string theory.

Role of Picard groups

The Picard group ( text{Pic}(V) ) of a variety ( V ) is the group of isomorphism classes of line bundles on ( V ), where the group operation is the tensor product. The Picard group helps to understand divisor classes more abstractly by classifying these classes via equivalence of line bundles.

Conclusion

Separators and intersections form the basis of the framework of algebraic geometry. Through these concepts, we can explore extensive features and symmetries of algebraic varieties. While initially abstract, they provide powerful tools for understanding complex geometry in practical and theoretical scenarios.

From linear equivalence and divisor classes to deep intersection products, these structures help mathematicians develop deeper insights into the framework of geometry, and lay the groundwork for further discoveries and innovations.


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