Algebraic Varieties
Introduction
An **algebraic variety** is a fundamental concept in algebraic geometry, a branch of mathematics that studies solutions to systems of polynomial equations. Algebraic varieties generalize the notion of algebraic curves and surfaces to higher dimensions and more complex structures. They play a crucial role in various areas of mathematics, including number theory, complex analysis, and mathematical physics.
Definition
An algebraic variety is defined as the set of solutions to a system of polynomial equations over a field. Formally, let \( K \) be a field, and let \( K[x_1, x_2, \ldots, x_n] \) be the ring of polynomials in \( n \) variables with coefficients in \( K \). An algebraic variety \( V \) in \( K^n \) is the set of common zeros of a finite set of polynomials \( \{ f_1, f_2, \ldots, f_m \} \subset K[x_1, x_2, \ldots, x_n] \). That is,
\[ V = \{ (a_1, a_2, \ldots, a_n) \in K^n \mid f_i(a_1, a_2, \ldots, a_n) = 0 \text{ for all } i \}. \]
Varieties can be classified into **affine varieties** and **projective varieties** depending on whether they are defined in affine space or projective space.
Affine Varieties
Affine varieties are subsets of affine space \( K^n \) defined as the common zeros of a set of polynomials. For example, the set of points \( (x, y) \in K^2 \) satisfying the equation \( x^2 + y^2 - 1 = 0 \) is an affine variety, specifically a circle in the plane.
Affine varieties can be studied using the tools of commutative algebra, particularly through the concept of the **coordinate ring**. The coordinate ring of an affine variety \( V \subset K^n \) is the quotient ring \( K[x_1, x_2, \ldots, x_n] / I(V) \), where \( I(V) \) is the ideal of polynomials that vanish on \( V \).
Projective Varieties
Projective varieties are defined in projective space \( \mathbb{P}^n(K) \), which is the set of lines through the origin in \( K^{n+1} \). A projective variety is the set of common zeros of a set of homogeneous polynomials. For example, the set of points \( [x:y:z] \in \mathbb{P}^2(K) \) satisfying the equation \( x^2 + y^2 - z^2 = 0 \) is a projective variety, specifically a conic in the projective plane.
Projective varieties are important because they exhibit better geometric properties than affine varieties, such as compactness. The study of projective varieties involves the use of **homogeneous coordinates** and **homogeneous ideals**.
Singular and Non-Singular Varieties
An algebraic variety is called **non-singular** or **smooth** if it does not have any singular points, where the tangent space is not well-defined. Otherwise, it is called **singular**. The singular points of a variety are the points where the Jacobian matrix of the defining polynomials does not have full rank.
Non-singular varieties have well-defined tangent spaces at every point, and their local behavior resembles that of Euclidean space. Singular varieties, on the other hand, can have more complicated local structures, such as cusps or nodes.
Dimension of Varieties
The **dimension** of an algebraic variety is a measure of its "size" or "complexity". Intuitively, the dimension of a variety is the number of independent parameters needed to describe points on the variety. For example, a curve has dimension 1, a surface has dimension 2, and so on.
Formally, the dimension of an affine variety \( V \) can be defined as the Krull dimension of its coordinate ring. For projective varieties, the dimension can be defined using the concept of **Hilbert polynomials**.
Irreducible Varieties
An algebraic variety is called **irreducible** if it cannot be expressed as the union of two proper subvarieties. Equivalently, a variety is irreducible if its defining ideal is a prime ideal. Irreducible varieties are the building blocks of algebraic geometry, as any variety can be decomposed into a finite union of irreducible components.
Morphisms and Rational Maps
A **morphism** between two algebraic varieties \( V \) and \( W \) is a map that is defined by polynomials. Morphisms preserve the algebraic structure of varieties and play a crucial role in the study of their properties. A **rational map** is a map that is defined by rational functions, which are ratios of polynomials.
Rational maps can be used to study the birational equivalence of varieties, which is an equivalence relation that identifies varieties that are "almost" the same, except for a lower-dimensional subset.
Examples of Algebraic Varieties
Curves
Algebraic curves are one-dimensional varieties. Examples include:
- **Elliptic Curves**: Defined by equations of the form \( y^2 = x^3 + ax + b \). These curves have applications in number theory and cryptography.
- **Hyperelliptic Curves**: Generalizations of elliptic curves, defined by equations of the form \( y^2 = f(x) \), where \( f(x) \) is a polynomial of degree greater than 3.
Surfaces
Algebraic surfaces are two-dimensional varieties. Examples include:
- **Quadric Surfaces**: Defined by quadratic equations in three variables, such as \( x^2 + y^2 + z^2 = 1 \).
- **K3 Surfaces**: A special class of surfaces with rich geometric and arithmetic properties.
Higher-Dimensional Varieties
Higher-dimensional varieties include:
- **Calabi-Yau Manifolds**: Important in string theory, defined by certain conditions on their Ricci curvature.
- **Grassmannians**: Varieties parameterizing linear subspaces of a fixed dimension in a vector space.
Applications of Algebraic Varieties
Algebraic varieties have numerous applications in mathematics and science, including:
- **Number Theory**: Varieties over finite fields and their zeta functions.
- **Cryptography**: Use of elliptic curves in public-key cryptography.
- **Physics**: Role of Calabi-Yau manifolds in string theory.
- **Coding Theory**: Algebraic geometry codes constructed from varieties.