1 Definition and basic properties

A regular map is a fundamental type of function in algebraic geometry. It is a map between algebraic varieties that is given locally by polynomial expressions, so it respects the algebraic structure of the objects involved. Regular maps provide the standard notion of an algebraic morphism in many classical settings, especially for affine and projective varieties.

Unlike an arbitrary set-theoretic function, a regular map must be defined everywhere on its domain and must be expressible in coordinates by algebraic formulas. This makes regular maps central to the study of geometric objects through their equations.

1.1 Informal description

Informally, a regular map is an algebraic map with no denominators or singular points on its domain. On an affine variety, each coordinate of the map is a polynomial function; on a projective variety, the coordinates are given by homogeneous polynomials arranged so that the map is well defined.

This notion captures the idea of a “natural” map in algebraic geometry. Projections, embeddings, and many parametrizations are regular, while maps involving division by a variable or formulas that break down at some points are not.

1.2 Formal definition

In modern language, a regular map between algebraic varieties is a morphism of varieties. The precise formulation depends on whether the varieties are affine or projective, but the local condition is the same: near every point, the map can be written using polynomial data.

1.2.1 Affine varieties

For affine varieties, a map \(f : X \to Y\) is regular if, after choosing affine coordinates on \(Y\), each coordinate function of \(f\) is the restriction of a polynomial function on \(X\). Equivalently, the map is induced by a homomorphism of coordinate rings in the opposite direction.

This definition ensures that regularity is an intrinsic algebraic property rather than merely a coordinate artifact.

1.2.2 Projective varieties

For projective varieties, regular maps are described using homogeneous coordinates. A map from a projective variety to another projective variety is regular if it can be given by homogeneous polynomials of the same degree, and if these polynomials do not vanish simultaneously at any point of the domain.

Because projective varieties are built from equivalence classes of homogeneous coordinates, the expressions defining a regular map must remain compatible with scalar rescaling.

1.3 Examples

A simple example is the coordinate projection from an affine space onto one of its factors. This is regular because each output coordinate is just one of the input coordinates.

Another example is the Veronese map, which sends a point to a list of monomials of a fixed degree. Such maps are regular and are often used to embed varieties into projective space.

On a smooth affine curve, a polynomial parametrization gives a regular map from the affine line whenever the polynomial expressions are defined everywhere on the line.

1.4 Nonexamples

A rational function with a denominator that can vanish on the domain is not a regular map on the whole domain. For instance, a formula such as \(x \mapsto 1/x\) fails to be regular at \(x=0\).

Similarly, a map that is algebraically defined on a dense open set but cannot be extended across a missing point is not regular on the whole variety. Such functions are usually treated as rational maps instead.

1.5 Basic properties

Regular maps are closed under composition. If two maps are regular, their composite is also regular. This makes algebraic varieties into a category with regular maps as morphisms.

They are also stable under restriction to open subsets: a regular map remains regular when viewed on a smaller open domain. In addition, regular maps behave well with respect to products, subvarieties, and embeddings.

2 Regular maps in affine algebraic geometry

Affine algebraic geometry offers the clearest setting for regular maps because coordinate rings provide an explicit algebraic description. In this context, regular maps correspond closely to ring homomorphisms and polynomial functions.

2.1 Coordinate ring interpretation

The coordinate ring of an affine variety encodes its polynomial functions. A regular map between affine varieties can be studied by examining how it acts on these rings.

2.1.1 Polynomial functions

On an affine variety, a polynomial function is obtained by restricting a polynomial in ambient affine space to the variety. A regular map is built from such functions coordinate by coordinate.

This viewpoint emphasizes that regular maps do not depend on analytic convergence or limiting processes. They are determined entirely by polynomial algebra.

2.1.2 Ring homomorphisms

A regular map \(f : X \to Y\) between affine varieties induces a ring homomorphism from the coordinate ring of \(Y\) to the coordinate ring of \(X\). This induced map sends each function on \(Y\) to its pullback along \(f\).

Conversely, suitable ring homomorphisms arise from regular maps. This duality is one of the key features of affine algebraic geometry.

2.2 Regular functions versus regular maps

Regular functions are scalar-valued regular maps. They are maps from a variety to the affine line, and they form a ring under pointwise addition and multiplication.

Regular maps generalize this idea to targets of arbitrary dimension. While a regular function is a single algebraic coordinate, a regular map consists of several such coordinates assembled into a vector-valued or geometric transformation.

2.3 Composition of regular maps

The composition of two regular maps is again regular because polynomial expressions remain polynomial after substitution. In affine coordinates, this is immediate: substituting polynomials into polynomials produces polynomials.

This closure under composition is essential for the categorical structure of algebraic varieties. It also allows regular maps to serve as the basic arrows in geometric constructions.

2.4 Restriction to open subsets

If a map is regular on a variety, its restriction to any open subvariety is also regular. This follows from the local nature of the definition.

Open subsets often arise when one removes a closed set where a rational expression might otherwise fail. Regularity on an open set is therefore a flexible but still algebraic notion.

3 Regular maps in projective algebraic geometry

Projective geometry introduces equivalence classes of homogeneous coordinates, so regular maps must be defined in a way that respects this projective structure. The homogeneous-polynomial description provides the correct framework.

3.1 Homogeneous coordinates

Projective space is described using homogeneous coordinates, where points are represented by nonzero tuples up to common scaling. A regular map between projective varieties is typically given by homogeneous polynomials of a fixed degree.

Because scaling a representative should not change the image, the defining polynomials must transform compatibly under rescaling of coordinates. This requirement is what makes projective regularity more delicate than affine regularity.

3.2 Well-definedness conditions

To be well defined, the homogeneous polynomials defining a projective map must not vanish simultaneously at any point of the domain. If they did, the image point would be undefined.

The polynomials must also yield the same projective point regardless of the chosen representative of the input. These conditions ensure that the map descends from coordinates to actual projective points.

3.3 Regular maps between projective varieties

A regular map between projective varieties can often be described by restricting a projective-space map to subvarieties. Such maps include embeddings of projective varieties, projections defined on appropriate domains, and morphisms induced by linear systems.

Many important constructions in projective geometry, including embeddings by ample line bundles in more advanced settings, are expressed through regular maps.

3.4 Rational maps and indeterminacy

A rational map is given by algebraic formulas on a dense open subset but may fail to be defined everywhere. The points where the formula breaks down form the indeterminacy locus.

Regular maps are rational maps with no indeterminacy on the domain. This distinction is especially important in projective geometry, where many naturally occurring maps are only rational until they are resolved or restricted.

4 Functorial viewpoint

From a modern perspective, regular maps are morphisms in a category of algebraic varieties. This viewpoint highlights their structural role and connects classical geometry with abstract algebra.

4.1 Morphisms of varieties

A regular map between varieties is the same as a morphism of varieties in the usual algebraic-geometric sense. The term “morphism” emphasizes that the map is compatible with the variety structure and not just with the underlying sets.

This interpretation unifies affine and projective cases under one concept, even though the concrete descriptions differ.

4.2 Categories of algebraic varieties

Algebraic varieties form a category whose objects are varieties and whose morphisms are regular maps. Identity maps are regular, and compositions of morphisms remain within the category.

This categorical organization allows geometric problems to be phrased in terms of universal properties, functorial constructions, and structural equivalences.

4.3 Equivalent formulations

Regular maps admit several equivalent descriptions, especially in the classical setting. These formulations are often used interchangeably depending on whether one prefers geometric, coordinate, or algebraic language.

4.3.1 Via coordinate rings

For affine varieties, a regular map corresponds contravariantly to a homomorphism of coordinate rings. This algebraic formulation is often the most efficient for proofs and computations.

It translates geometric questions into questions about polynomial algebras and ideals.

4.3.2 Via local polynomial expressions

Geometrically, a regular map is one that can be written locally by polynomial formulas. This local description is especially useful for understanding how regularity behaves under restriction, gluing, and change of coordinates.

The equivalence of the local and algebraic definitions is a major organizing principle in the subject.

Several closely related notions help distinguish regular maps from other kinds of algebraic or geometric functions. Each captures a different level of generality or flexibility.

5.1 Regular functions

Regular functions are regular maps with target affine line. They are the simplest algebraic functions on a variety and form a ring.

They are often used to study the global geometry of a variety through its functions.

5.2 Rational functions

Rational functions are quotients of polynomial functions that may fail to be defined where the denominator vanishes. They are more general than regular functions and often more general than regular maps.

In many geometric problems, rational functions provide a useful intermediate step before asking whether a construction extends to a regular map.

5.3 Morphisms of schemes

In scheme theory, regular maps are replaced by morphisms of schemes, which extend the classical notion to much broader settings. For classical varieties, the scheme-theoretic morphism agrees with the usual regular map.

This framework allows algebraic geometry to treat singularities, arithmetic properties, and more general base rings in a unified way.

5.4 Polynomial maps

Polynomial maps are maps between affine spaces or varieties given by polynomial formulas. They are a concrete special case of regular maps.

In affine settings, the distinction between polynomial maps and regular maps is often small, but regular maps are the more intrinsic concept because they are defined on varieties rather than only on ambient coordinate spaces.

6 Applications and uses

Regular maps are used throughout algebraic geometry because they provide the basic language for constructing, comparing, and classifying varieties. They also appear in related areas such as commutative algebra and geometric modeling.

6.1 Algebraic geometry

In algebraic geometry, regular maps are the standard notion of a geometric transformation. They are used to define isomorphisms, embeddings, projections, fibrations, and many other structural maps.

Their algebraic nature allows geometric properties to be studied through equations and coordinate rings.

6.2 Classification problems

Classification problems often rely on comparing varieties up to regular isomorphism. Two varieties are considered the same in the strongest algebraic sense when there is a regular map with a regular inverse between them.

Regular maps also help detect invariants preserved under algebraic equivalence, making them essential tools in organizing families of varieties.

6.3 Parametrization of varieties

Regular maps are used to parametrize algebraic varieties by simpler spaces such as affine or projective spaces. Such parametrizations can reveal hidden structure and simplify computation.

They are especially useful when a variety can be described as the image of a regular map from a space with known coordinates. In favorable cases, a regular parametrization can turn geometric questions into explicit algebraic ones.