1 Definition and basic examples

An affine variety is the solution set of one or more polynomial equations in an affine space over a field. It is a basic object in algebraic geometry and provides a natural way to study algebraic relations among coordinates using geometric language. Affine varieties appear whenever one asks for the common zero set of polynomials and then analyzes the resulting shape, singularities, and algebraic structure.

1.1 Affine space and polynomial equations

Affine space of dimension n over a field consists of points with n coordinates, usually written as K^n. Unlike vector space terminology, affine space emphasizes points rather than vectors, although it is modeled on a vector space. Polynomial equations in these coordinates define constraints whose simultaneous solutions are the main objects of study.

A single polynomial equation may define a curve, surface, or higher-dimensional set, depending on the number of variables and the field. Multiple equations are typically considered together, since their common solution set often has more refined structure than any one equation alone.

1.2 Algebraic sets in affine space

An algebraic set is any subset of affine space defined as the common zero locus of a collection of polynomials. These sets form the foundational closed sets in the Zariski topology and include both reducible and irreducible examples. Affine varieties are algebraic sets with an additional irreducibility condition in the classical sense.

1.2.1 Zero loci of polynomial ideals

Because only the ideal generated by a collection of polynomials matters for its common zeros, algebraic sets are naturally associated with ideals in a polynomial ring. The zero locus of an ideal consists of all points where every polynomial in the ideal vanishes. This viewpoint makes the correspondence between geometry and algebra especially transparent.

1.2.2 Simple examples in low dimensions

In one affine coordinate, the zero set of a nonzero polynomial over an algebraically closed field is usually finite. In two coordinates, equations such as y - x^2 define familiar curves like parabolas. Systems like x = 0 and y = 0 define a single point, while x y = 0 in the plane gives the union of two coordinate axes, illustrating reducible algebraic sets.

1.3 Affine varieties as irreducible algebraic sets

Classically, an affine variety is often defined as an irreducible algebraic set, meaning it cannot be expressed as the union of two proper algebraic subsets. Irreducibility corresponds to the idea that the space is geometrically connected in an algebraic sense, though not necessarily topologically connected in the usual topology. This definition aligns well with the study of prime ideals and function fields.

2 Coordinate rings and ideals

The algebraic study of affine varieties relies on the coordinate ring, which records polynomial functions on the variety. Ideals determine the defining equations, while the ring structure captures how functions behave on the set. This interplay is one of the central themes of commutative algebraic geometry.

2.1 Polynomial rings and vanishing ideals

For affine space K^n, the polynomial ring K[x_1, ..., x_n] contains all polynomial expressions in the coordinates. Given a subset of affine space, its vanishing ideal consists of all polynomials that vanish at every point of the set. This ideal encapsulates the equations satisfied by the subset.

2.2 The coordinate ring of an affine variety

The coordinate ring is the ring of polynomial functions on the variety, obtained by identifying polynomials that agree on the variety. It serves as the algebraic counterpart of the geometric space and is often the main object used in calculations. Many geometric properties can be read from this ring.

2.2.1 Quotient by the ideal of the variety

If V is an affine variety with vanishing ideal I(V), then its coordinate ring is K[x_1, ..., x_n]/I(V). Elements of this quotient correspond to polynomial functions restricted to V. Two polynomials represent the same function on V precisely when their difference lies in I(V).

2.2.2 Relation to functions on the variety

Polynomial functions are regular functions on affine varieties, and the coordinate ring captures all of them. This makes affine varieties especially tractable, because their global regular functions are algebraic rather than merely set-theoretic. In many cases, geometric maps between varieties can be described entirely through ring homomorphisms.

2.3 Radical and prime ideals

The ideal associated with an algebraic set is radical, reflecting the fact that only the set of common zeros matters, not multiplicities. Prime ideals correspond to irreducible algebraic sets and therefore to affine varieties in the classical sense. These ideal-theoretic distinctions are crucial for understanding the geometry encoded by polynomial equations.

2.3.1 Nullstellensatz correspondence

Hilbert’s Nullstellensatz establishes a fundamental link between ideals and algebraic sets over algebraically closed fields. Roughly, it states that the ideal of all polynomials vanishing on a zero locus is the radical of the original ideal. This result provides the backbone of the dictionary between geometric sets and algebraic ideals.

2.3.2 Irreducibility and primeness

An algebraic set is irreducible if and only if its vanishing ideal is prime. Prime ideals ensure that the coordinate ring has no zero divisors, mirroring the impossibility of decomposing the variety into smaller closed pieces. This equivalence makes primeness one of the most important algebraic criteria in the subject.

3 Morphisms of affine varieties

Morphisms are the natural maps between affine varieties. They preserve the algebraic structure by being given coordinatewise by polynomial expressions, and they correspond contravariantly to homomorphisms of coordinate rings. This duality is a central organizing principle in affine algebraic geometry.

3.1 Regular maps

A regular map between affine varieties is one whose coordinate functions are polynomial functions on the source. Such maps are the algebraic analogues of smooth or continuous maps, but with stronger rigidity. They are exactly the maps compatible with the polynomial structure of the varieties.

3.1.1 Polynomial functions on varieties

On an affine variety, a polynomial in ambient coordinates defines a function by evaluation at each point. Different polynomials may define the same function if they agree on the variety. Regular maps are built from these functions, ensuring that the map is algebraic in each coordinate.

3.1.2 Coordinate descriptions of maps

A map from one affine variety to another can often be written as a tuple of polynomial functions. For example, a map into affine m-space is determined by m coordinate functions. This description makes composition straightforward and connects directly with ring homomorphisms in the opposite direction.

3.2 Isomorphisms and automorphisms

An isomorphism of affine varieties is a regular map with a regular inverse. It identifies two varieties as the same object from the algebraic geometric viewpoint. An automorphism is an isomorphism from a variety to itself and reflects its internal symmetries.

3.3 Products and projections

The product of affine varieties is again affine, with coordinates formed by combining those of the factors. Projections onto factors are regular maps and often serve as basic examples of morphisms. Product constructions are useful for describing families of varieties and multi-parameter algebraic systems.

4 Geometric properties

Affine varieties exhibit geometric features such as dimension, smoothness, and tangent behavior, all of which can be studied algebraically. These properties often reveal how equations constrain the space locally and globally. They also distinguish well-behaved varieties from those with special points or decompositions.

4.1 Dimension

Dimension measures the size of an affine variety in an algebraic sense. It can be defined using chains of irreducible closed subsets or via the associated coordinate ring. This notion agrees with geometric intuition in familiar examples.

4.1.1 Krull dimension

The Krull dimension of the coordinate ring is the maximal length of a chain of prime ideals, and it matches the dimension of the variety. This algebraic definition is especially useful because it is computable and stable under many constructions. It also connects naturally to the behavior of subvarieties and components.

4.1.2 Intuitive geometric meaning

A curve has dimension one, a surface has dimension two, and affine n-space has dimension n. More generally, dimension counts the number of independent parameters needed to describe a generic point of the variety. Singularities and reducible components may complicate local appearance without changing the overall dimension.

4.2 Smooth and singular points

A point on an affine variety is smooth if the variety looks locally like a space of the expected dimension near that point. Singular points are those where this local picture fails. The distinction is important in both local geometry and broader structural analysis.

4.2.1 Jacobian criterion

For varieties defined by equations, smoothness can often be tested using the Jacobian matrix of partial derivatives. The rank of this matrix at a point determines whether the local dimension behaves as expected. This criterion provides a practical method for locating singularities.

4.2.2 Examples of singular affine varieties

The curve defined by y^2 = x^3 has a cusp at the origin, while y^2 = x^2(x + 1) has a node-like singularity at the origin. Such examples show how algebraic equations can produce points where tangents are not well behaved. Singularities are central objects of study because they often control the local complexity of a variety.

4.3 Tangent spaces

The tangent space at a smooth point is a linear approximation to the variety near that point. It can be defined using derivatives of the defining equations or through derivations on the local ring. Tangent spaces help measure local dimension and detect whether the variety is smooth.

5 Structure theory

The structure of an affine variety is reflected in how it decomposes into simpler pieces and how its closed subsets fit together. Algebraic methods describe these features using ideals, topology, and decomposition theorems. This part of the theory clarifies how complicated varieties are built from irreducible components.

5.1 Irreducible decomposition

Every algebraic set can be written uniquely as a finite union of irreducible components, none contained in another. This decomposition parallels factorization into fundamental pieces. It is often the first step in analyzing the geometry of a variety.

5.2 Closed subvarieties

A closed subvariety is a closed algebraic subset of a variety, defined by adding more polynomial equations. Such subvarieties inherit much of the ambient structure while reflecting additional constraints. They play a central role in studying intersections, dimension drops, and chains of inclusions.

5.3 Zariski topology

The Zariski topology is the natural topology on affine varieties in which algebraic sets are closed. It is coarse compared with familiar Euclidean topology, but it is well suited to algebraic questions. Many geometric notions become simpler and more algebraic in this setting.

5.3.1 Closed sets and basic opens

Closed sets in the Zariski topology are zero loci of polynomial families, while basic open sets are complements of such loci. These opens are often large and dense, reflecting the rigid nature of algebraic constraints. They provide the right setting for local algebraic arguments.

5.3.2 Density and generic points

A subset is dense if its closure is the whole variety in the Zariski topology. This often means that a property holds on a nonempty open set and therefore generically. Generic points capture the idea of a typical point of an irreducible component, even when no single ordinary point fully represents it.

6 Classical examples

Classical examples make the abstract definitions concrete and illustrate the range of possible shapes. They show how familiar curves and surfaces arise as affine varieties and how singularities or decompositions appear in practice. These examples are often the first models studied in algebraic geometry.

6.1 Affine lines and planes

The affine line is the simplest variety and corresponds to a single coordinate axis. The affine plane is the next basic case and serves as the ambient space for many curves. These objects are the setting in which polynomial equations become visible as geometric figures.

6.2 Conics and quadrics

Conics are plane curves defined by quadratic equations, such as ellipses, parabolas, and hyperbolas in suitable fields and coordinate systems. Quadrics extend this idea to higher dimensions, including surfaces defined by second-degree polynomials. Their classification is a classical topic with deep links to linear algebra.

6.3 Affine curves

Affine curves are one-dimensional affine varieties. They include many of the most studied examples in algebraic geometry because they are rich enough to exhibit interesting behavior while remaining relatively manageable. Their coordinate rings often provide accessible examples of the general theory.

6.3.1 Parabolas and cubics

A parabola is given by an equation such as y = x^2, while cubic curves may have more complicated shapes and singularities. Cubics are especially significant because they can show both smooth and singular behavior depending on coefficients. Their algebraic properties are closely tied to the geometry of their points.

6.3.2 Singular plane curves

Singular plane curves include cusps and nodes, where the curve fails to be smooth. These examples are important because they illustrate how local geometry can differ sharply from the behavior of nearby smooth points. They also serve as test cases for tangent spaces and the Jacobian criterion.

6.4 Affine surfaces

Affine surfaces are two-dimensional affine varieties and often arise as zero sets of one equation in affine three-space or as intersections of several equations. They can display rich geometry, including smooth regions, singular points, and multiple components. Their study lies between the relative simplicity of curves and the complexity of higher-dimensional varieties.

7 Relation to projective varieties

Affine varieties are closely related to projective varieties, which are studied in projective space. Many projective objects can be examined through affine pieces, and affine methods often simplify local analysis. The relationship between the two settings is a major theme in algebraic geometry.

7.1 Affine patches of projective varieties

A projective variety can be covered by affine patches, each obtained by setting one homogeneous coordinate to a nonzero value. These patches allow global projective questions to be studied locally using affine techniques. Many local properties of a projective variety are visible in such coordinate charts.

7.2 Homogenization and dehomogenization

Homogenization turns an affine polynomial into a homogeneous polynomial by adding a new variable, producing a projective equation. Dehomogenization reverses this process on a chosen affine chart. Together, these procedures link affine equations with projective closures.

7.3 Compactification ideas

Projective space provides a way to compactify affine varieties by adding points at infinity. This often clarifies global behavior, such as how curves intersect the boundary of an affine chart. Compactification is useful for studying completeness, intersections, and asymptotic geometry.

8 Applications and significance

Affine varieties are fundamental not only in pure algebraic geometry but also in computation and in the structural study of polynomial equations. They provide a common language for geometry, algebra, and algorithmic methods. Their importance comes from both theoretical depth and practical versatility.

8.1 Solving polynomial systems

Systems of polynomial equations can be interpreted as affine varieties, making geometric methods available for their study. Questions about existence, number, and structure of solutions become questions about algebraic sets and coordinate rings. This perspective underlies elimination theory and many symbolic techniques.

8.2 Commutative algebra and algebraic geometry

Affine varieties form a bridge between commutative algebra and geometry. Ideals, prime decomposition, radicals, and quotient rings translate naturally into geometric language. Conversely, geometric properties such as dimension and irreducibility motivate algebraic invariants.

Computational algebraic geometry develops algorithms for manipulating ideals and varieties, including Gröbner bases and elimination procedures. These methods make it possible to compute zero loci, test ideal membership, and analyze dimensions in concrete cases. Affine varieties are among the primary objects these algorithms are designed to handle.