1 Basic definition

A rational map is a map between algebraic varieties that is given by algebraic formulas on a large open part of the source. In practice, it is described by quotients of polynomial functions, so it may be undefined where denominators vanish. Rational maps are one of the standard tools for describing geometric transformations in algebraic geometry.

1.1 Rational functions and maps

A rational function is a quotient of two polynomials, or more generally an element of the function field of a variety. A rational map generalizes this idea from functions on a single variety to maps between varieties. Each coordinate of the map is represented by a rational function, so the map is algebraic wherever all expressions are defined.

1.2 Domain of definition

The domain of definition is the set of points where the formulas for a rational map make sense. This domain is usually a dense open subset of the source variety. Dense openness reflects the fact that the map is determined by its behavior on a large region, even if it is not everywhere defined.

1.3 Indeterminacy locus

The indeterminacy locus is the subset where a rational map fails to be defined. It is typically a smaller closed set, often of high codimension. At such points, the coordinate expressions may become undefined because all available representatives vanish simultaneously or because the quotient cannot be evaluated consistently.

1.4 Examples

A simple example is the map on the affine plane given by \((x,y) \mapsto (x/y, y)\), which is defined wherever \(y \neq 0\). Another common example appears in projective geometry, where coordinate ratios define maps on projective space except at points where all defining expressions vanish. Such examples illustrate both the flexibility and the partial nature of rational maps.

2 Rational maps in algebraic geometry

Rational maps play a central role in algebraic geometry because many natural geometric constructions are not globally regular but are still algebraically meaningful on large subsets. They are especially important for comparing varieties, studying function fields, and describing transformations that are invertible only on dense open sets.

2.1 Maps between affine varieties

For affine varieties, a rational map can often be written using polynomial fractions in affine coordinates. The map is defined on the points where the denominators do not vanish. Affine varieties provide a convenient setting for concrete calculations, since rational maps can be handled with explicit coordinate expressions.

2.2 Maps between projective varieties

On projective varieties, rational maps are often given by homogeneous polynomial data. Because projective points are equivalence classes of nonzero coordinates, the formulas must respect scaling. This makes projective rational maps more subtle than affine ones, but also more natural in many geometric problems.

2.3 Homogeneous coordinates

Homogeneous coordinates provide a coordinate system suited to projective varieties. A rational map in projective space is typically represented by a collection of homogeneous polynomials of the same degree. The use of homogeneous coordinates ensures that the resulting point is well defined up to overall scaling.

2.4 Regular maps versus rational maps

A regular map is defined everywhere on its source and is given by polynomial expressions in local coordinates. A rational map may fail to be defined on a closed subset, but agrees with a regular map on its domain of definition. Every regular map is rational, while not every rational map is regular.

3 Properties and classification

Rational maps are classified by how much of the target they reach, whether they can be inverted, and how they behave under composition. These properties connect the study of rational maps with function fields and with the geometry of varieties themselves.

3.1 Equality of rational maps

Two rational maps are considered equal if they agree on a dense open subset where both are defined. This notion reflects the idea that a rational map is determined by its generic behavior rather than by values at exceptional points. Equality in this sense is compatible with the algebraic nature of the map.

3.2 Composition of rational maps

The composition of rational maps is again rational whenever the image of the first map meets the domain of definition of the second in a sufficiently large set. In general, care is needed because indeterminacy may propagate through composition. Nonetheless, composition is a fundamental operation and is essential for dynamical applications.

3.3 Dominant rational maps

A rational map is dominant if its image is dense in the target variety. Dominance indicates that the map is large enough to capture the generic points of the target. Such maps are important because they induce embeddings of function fields and often reflect deep geometric relations between varieties.

3.4 Birational maps

A birational map is a rational map that has a rational inverse. Birational maps identify varieties that are equivalent from the viewpoint of their function fields and their generic geometry. They form the basis of birational geometry, where one studies varieties up to transformations that are invertible almost everywhere.

3.4.1 Birational equivalence

Two varieties are birationally equivalent if there exists a birational map between them. This relation is an equivalence relation on varieties of the same dimension. Birational equivalence preserves many generic properties while allowing singular or exceptional loci to vary.

3.4.2 Inverse rational maps

The inverse of a birational map is itself rational, though it may have its own indeterminacy locus. Constructing inverse rational maps is often easier on dense open subsets than on the whole variety. Their existence shows that birational maps are reversible in the generic algebraic sense.

4 Resolution of indeterminacy

A central theme in the theory of rational maps is replacing a partially defined map by a well-defined morphism after modifying the source. Resolution of indeterminacy makes it possible to study rational maps using tools from regular morphisms and smooth geometry.

4.1 Blowing up

Blowing up is a geometric operation that replaces a subvariety by a higher-dimensional exceptional divisor. It is frequently used to separate directions that cause a rational map to fail at a point or along a subvariety. After enough blowups, an indeterminate map may become regular.

4.2 Proper transforms

The proper transform of a subvariety is its modified image under a blowup. Proper transforms help track how geometric objects change when singularities or base points are resolved. They provide a precise way to compare the original variety with its altered version.

4.3 Elimination of base points

Base points are points where all defining expressions of a rational map vanish simultaneously. Eliminating base points is often equivalent to resolving indeterminacy. The process is essential in both theoretical work and explicit computations, since it converts a partially defined map into a genuine morphism.

5 Rational maps in projective space

Projective space is one of the most important settings for rational maps, since many geometric transformations are naturally homogeneous. In this context, rational maps are described by polynomial tuples and are studied through their base loci and degrees.

5.1 Homogeneous polynomial representations

A rational map on projective space is commonly given by homogeneous polynomials of the same degree. Such a representation defines a map wherever the polynomials do not vanish simultaneously. Different polynomial tuples may represent the same rational map if they differ by a common nonzero factor.

5.2 Base loci

The base locus is the set of points where all defining homogeneous polynomials vanish. It coincides with the indeterminacy locus for many projective rational maps. Understanding the base locus is crucial for analyzing where the map is defined and how it behaves near exceptional points.

5.3 Degree of a rational map

The degree of a rational map in projective space measures the common degree of its defining homogeneous polynomials, after removing common factors when appropriate. Degree provides a coarse but useful invariant of complexity. Under composition, degrees often grow, which is especially important in dynamics.

6 Rational dynamics

When a rational map is iterated, the repeated action can produce intricate dynamical patterns. Rational dynamics studies the long-term behavior of points under successive application of the same map and connects algebraic geometry with complex and real dynamical systems.

6.1 Iteration of rational maps

Iteration means composing a rational map with itself many times. Each iterate may have a larger indeterminacy set or a more complicated algebraic form. Iteration is a natural way to probe stability, growth, and recurring geometric structures.

6.2 Fixed points and periodic points

A fixed point is a point that maps to itself, while a periodic point returns to itself after finitely many iterations. These points organize much of the dynamical structure of a rational map. Their local behavior can reveal whether nearby orbits are attracted, repelled, or neutral.

6.3 Julia sets and Fatou sets

In complex dynamics, the Julia set is the locus of chaotic or unstable behavior, while the Fatou set is the region of stable dynamics. For rational maps, these sets partition the space into contrasting dynamical regimes. Their study has become a major part of one-dimensional complex dynamics and related higher-dimensional theories.

6.4 Stability and bifurcation

Stability concerns how the qualitative behavior of a rational map changes under small perturbations. Bifurcation refers to a change in dynamical structure as parameters vary. These ideas are central in understanding families of rational maps, where slight algebraic changes can produce different orbit patterns.

7 Applications

Rational maps appear in many branches of mathematics and in computational settings where algebraic formulas are used to model geometric transformations. Their partial definability makes them especially useful for describing processes that are natural on generic points but singular at special configurations.

7.1 Algebraic geometry

In algebraic geometry, rational maps are used to compare varieties, study function fields, and classify geometric objects up to birational equivalence. They are indispensable in modern treatments of surfaces, higher-dimensional varieties, and moduli problems. Many key constructions are phrased most naturally in rational terms.

7.2 Computer algebra

Computer algebra systems often manipulate rational maps symbolically through polynomial arithmetic and elimination methods. Such computations arise in solving equations, simplifying expressions, and studying geometric configurations. Rational maps also appear in algorithmic approaches to elimination theory and implicitization.

7.3 Dynamical systems

In dynamical systems, rational maps provide explicit nonlinear models with rich iterative behavior. They are studied over the real or complex numbers and are used to analyze stability, chaos, and orbit structure. Their algebraic form makes them accessible to both symbolic and numerical methods.

7.4 Kinematics and robotics

Rational maps are used to describe motions and constraints in kinematics and robotics. Many position and orientation relations can be encoded algebraically, especially after suitable parameterization. Rational expressions help model linkage mechanisms, inverse problems, and configuration spaces.

Several closely related notions help place rational maps within algebraic geometry and complex analysis. These concepts clarify the relationship between algebraic formulas, globally defined maps, and analytic generalizations.

8.1 Rational functions

Rational functions are quotients of polynomials or elements of a function field. They are the scalar building blocks from which rational maps are formed. Understanding rational functions is essential for working with coordinate descriptions of varieties.

8.2 Morphisms

A morphism is a globally defined regular map between algebraic varieties. Morphisms are the everywhere-defined counterpart of rational maps. Many rational maps become morphisms after resolving indeterminacy or restricting to a suitable open subset.

8.3 Meromorphic maps

Meromorphic maps are analytic analogues of rational maps, especially in complex geometry. They are locally given by quotients of holomorphic functions. The comparison between meromorphic and rational maps helps bridge algebraic and analytic viewpoints.

8.4 Cremona transformations

Cremona transformations are birational maps of projective space. They are classical examples of rational maps with rich algebraic structure. Their study has influenced birational geometry, projective geometry, and the theory of polynomial automorphisms.