1 Definitions and basic examples
An algebraic variety is a geometric object defined by polynomial equations. In its most familiar form, it is the set of common zeros of one or more polynomials, studied together with the algebraic data needed to describe functions on it. This viewpoint makes it possible to translate geometric questions into algebraic ones and vice versa.
In modern usage, the term may be applied in several related ways. It can mean an affine variety, a projective variety, or a more general space equipped with a structure sheaf. Despite these differences, the central idea remains the same: geometry is encoded by polynomial relations.
1.1 Polynomial equations and zero sets
A polynomial equation in several variables defines a locus where the polynomial vanishes. A system of such equations determines the set of points satisfying all of them simultaneously. These sets are the basic building blocks of algebraic geometry.
The ambient space may be ordinary affine space, where coordinates are unrestricted, or projective space, where points are considered up to scaling. The choice of setting affects both the geometry and the kinds of tools used to study the object.
1.2 Affine algebraic varieties
An affine algebraic variety is typically defined as the common zero set of a collection of polynomials in affine space. Classical examples include lines, curves, and surfaces described by polynomial equations in coordinates.
Affine varieties are closely tied to polynomial functions on them. Their geometry is reflected in the algebra of functions that remain well defined on the entire set, making them especially suitable for algebraic analysis.
1.3 Projective algebraic varieties
A projective algebraic variety is defined in projective space by homogeneous polynomial equations. Projective space is useful because it adds points at infinity, often simplifying geometric statements and ensuring that certain intersections behave more regularly.
Projective varieties are central in many parts of algebraic geometry. They provide a natural setting for compactness-like properties and for studying curves and surfaces in a unified framework.
1.4 Examples
Many familiar geometric figures can be expressed as algebraic varieties. Even simple shapes often arise from low-degree polynomial equations, which makes them accessible entry points into the subject.
1.4.1 Lines and conics
A line is the simplest algebraic variety, defined by a linear equation. Conics, such as circles, ellipses, parabolas, and hyperbolas, arise from quadratic equations. These examples illustrate how the degree of the defining polynomial influences the shape of the variety.
1.4.2 Curves and surfaces
More complicated examples include algebraic curves and surfaces, such as cubic curves or quadratic surfaces. Their geometry may range from smooth and well behaved to highly intricate, depending on the defining equations and their intersections.
1.5 Irreducibility and components
An algebraic variety may sometimes decompose into simpler pieces. If it cannot be written as a union of two proper closed algebraic subsets, it is called irreducible. Otherwise, it has several irreducible components.
This decomposition helps organize the geometry of a variety. Studying each component separately often reveals the structure of the whole object more clearly.
2 Algebraic structure
The geometry of a variety is controlled by algebraic data associated with polynomial functions. This connection is one of the main reasons algebraic varieties are so powerful: geometric features can often be read from rings and ideals.
2.1 Coordinate rings
For an affine variety, the coordinate ring is the ring of polynomial functions restricted to the variety. It records how polynomials behave on the set of points and serves as an algebraic model of the geometry.
Properties of the coordinate ring often mirror geometric properties of the variety. For example, the absence of zero divisors is related to irreducibility in the affine case.
2.2 Ideals and the Nullstellensatz
The set of polynomials vanishing on a variety forms an ideal. Conversely, an ideal in a polynomial ring defines a zero set. The correspondence between ideals and algebraic sets is a central theme of the subject.
Hilbert’s Nullstellensatz gives a precise relationship between polynomial ideals and their common zero sets over algebraically closed fields. It is a foundational theorem that links algebraic and geometric reasoning.
2.3 Morphisms between varieties
A morphism is a map between varieties defined by polynomials. Such maps preserve the algebraic structure and are the natural notion of function in algebraic geometry.
Morphisms allow varieties to be compared, combined, and classified. They include projections, inclusions, and more elaborate maps arising from polynomial formulas.
2.4 Isomorphisms and embeddings
An isomorphism is a morphism with an inverse that is also a morphism. Two isomorphic varieties are regarded as having the same algebraic-geometric structure, even if they appear different in coordinates.
An embedding places one variety inside another in a way that preserves its structure. Embeddings are useful for representing abstract varieties concretely within affine or projective space.
3 Geometric properties
Beyond their defining equations, varieties have intrinsic geometric characteristics. These include size, local shape, and the behavior of points where the object is not smooth.
3.1 Dimension
The dimension of a variety measures its geometric complexity. A curve has dimension one, a surface has dimension two, and higher-dimensional varieties extend this pattern.
Dimension can be defined in several equivalent ways, including through chains of subvarieties or the behavior of coordinate rings. It is one of the most fundamental invariants in the subject.
3.2 Smoothness and singularities
A smooth point is one where the variety looks locally like ordinary space of the expected dimension. Singularities are points where this regular behavior fails.
Singularities may arise from cusps, crossings, or more complicated local structures. Their study is a major area of algebraic geometry because singular points often control global properties of a variety.
3.3 Tangent spaces
The tangent space at a point captures the first-order behavior of a variety near that point. For a smooth point, it approximates the variety by a linear space.
Tangent spaces are useful for understanding local geometry, including dimension and smoothness. They also provide a bridge between algebraic equations and differential intuition.
3.4 Degree
The degree of a projective variety measures, in a broad sense, how it intersects linear spaces of complementary dimension. For curves and surfaces, it is closely related to the degree of the defining equations.
Degree is an important numerical invariant. It often reflects both complexity and intersection behavior, making it a standard tool in classification and comparison.
4 Classification and special types
Varieties are often grouped by dimension, defining equations, or geometric behavior. Some classes have especially rich theories and recur frequently across algebraic geometry.
4.1 Curves
An algebraic curve is a one-dimensional variety. Curves are among the most thoroughly studied objects in the field, with deep connections to function theory, topology, and arithmetic.
Their classification depends on invariants such as genus, singularities, and projective model. Even simple curves can display subtle global behavior.
4.2 Surfaces
An algebraic surface is a two-dimensional variety. Surfaces are more complex than curves and admit a wide range of phenomena, including intricate singularities and birational transformations.
The study of surfaces plays a major role in classification theory. Many central ideas in higher-dimensional geometry first appear in this setting.
4.3 Hypersurfaces
A hypersurface is defined by a single polynomial equation in a larger ambient space. Examples include quadric surfaces and cubic hypersurfaces.
Because they are cut out by one equation, hypersurfaces are often easier to describe than more complicated varieties. At the same time, they can exhibit rich geometry and singular behavior.
4.4 Complete intersections
A complete intersection is a variety defined by as many equations as its codimension suggests in a suitable ambient space. These varieties are frequently more tractable than arbitrary intersections.
Complete intersections occupy an important middle ground between simplicity and complexity. They arise naturally in both theoretical and computational contexts.
5 Construction and operations
Varieties can be built from simpler ones and combined through geometric operations. These constructions help extend the theory and produce new examples.
5.1 Subvarieties
A subvariety is a variety contained inside another variety, usually defined by adding more polynomial equations. Subvarieties allow one to study smaller geometric pieces within a larger space.
They are essential for understanding dimension, decomposition, and local structure. Many important objects in algebraic geometry are naturally described as subvarieties of projective space or other ambient varieties.
5.2 Products of varieties
The product of two varieties combines them into a new variety whose points are pairs of points from the factors. This construction is the algebraic-geometric analogue of forming Cartesian products.
Products are useful for building higher-dimensional examples and for formulating geometric questions involving multiple variables or parameters.
5.3 Fiber products
A fiber product is a refined construction that combines varieties over a common base. It captures how two maps to the same target interact.
This operation is fundamental in modern algebraic geometry because it preserves information about families, base change, and intersections in a flexible way.
5.4 Blowups
A blowup is a transformation that replaces a subvariety with a larger geometric object that records directional information near the chosen locus. It is often used to simplify singularities or study birational geometry.
Blowups are central tools in resolving singular points and understanding how varieties change under controlled modifications.
6 Advanced topics
Advanced study of varieties involves comparing them up to birational equivalence, analyzing their function fields, and extending the framework to more general geometric objects.
6.1 Birational equivalence
Two varieties are birationally equivalent if they agree on dense open subsets. This relation identifies objects that have the same rational geometry.
Birational equivalence is weaker than isomorphism but still captures many essential features. It is especially important in classification problems.
6.2 Rationality
A variety is rational if it is birationally equivalent to projective or affine space. Rational varieties are, in a sense, the simplest from the viewpoint of function fields.
Determining whether a variety is rational can be highly nontrivial. The problem connects geometry with arithmetic and invariant theory.
6.3 Divisors and line bundles
Divisors formalize codimension-one subvarieties and their intersections, while line bundles encode families of one-dimensional vector spaces varying over a variety. Together they provide a language for studying global geometry.
These notions are fundamental in the theory of curves, surfaces, and higher-dimensional varieties. They also play a key role in intersection theory and classification.
6.4 Sheaves and schemes
Sheaves organize local algebraic information and describe how local data are glued together across a variety. Schemes generalize varieties by allowing more flexible underlying rings and by incorporating nilpotent elements.
The scheme-theoretic viewpoint is now standard in modern algebraic geometry. It unifies classical varieties with broader arithmetic and geometric constructions.
7 Applications
Algebraic varieties appear in many areas of mathematics and related fields. Their utility comes from the ability to encode complex structure in polynomial form.
7.1 Number theory
Varieties are used to study solutions of polynomial equations with integer or rational coordinates. This perspective lies at the heart of Diophantine geometry.
Questions about rational points, local-global principles, and reduction modulo primes often depend on the geometry of varieties. In this way, geometric methods inform arithmetic problems.
7.2 Physics
Algebraic varieties appear in several areas of theoretical physics, especially where symmetry and constrained systems are important. They are used in the study of moduli spaces, string theory, and certain models of phase structure.
Polynomial geometry also helps describe parameter spaces and conserved quantities. The algebraic viewpoint can make complex physical relationships more manageable.
7.3 Computer algebra
Varieties play a major role in symbolic computation. Algorithms for solving polynomial systems, eliminating variables, and studying solution sets are based on algebraic-geometric ideas.
Computational methods are used in robotics, coding theory, optimization, and automated theorem proving. The geometric language helps organize these procedures conceptually.
7.4 Cryptography
Some cryptographic systems use the arithmetic of algebraic varieties, particularly those of elliptic curves and related objects. The structure of rational points on these varieties can provide hard computational problems.
This application has made algebraic geometry relevant to modern secure communication. The underlying theory combines geometry, arithmetic, and algorithmic practice.
8 History
The theory of algebraic varieties developed over many centuries, beginning with classical studies of curves and surfaces and evolving into a highly abstract modern discipline.
8.1 Classical origins
Early work on polynomial equations led to the study of geometric loci, especially conics and algebraic curves. Mathematicians gradually recognized that equations could define intricate shapes with rich properties.
During the nineteenth century, foundational ideas in geometry and invariant theory helped shape the classical theory of varieties. The subject became increasingly systematic as methods for handling singularities and intersections improved.
8.2 Modern development
In the twentieth century, algebraic geometry was transformed by the introduction of abstract varieties, sheaf theory, and schemes. These tools extended the scope of the subject and clarified many earlier results.
Modern development also connected algebraic varieties to topology, number theory, and representation theory. As a result, varieties became central objects in a broad network of mathematical ideas.