Glossary

8.1 Rational maps between varieties

3 min read•july 30, 2024

Rational maps between varieties are like secret passages in a maze. They're not always open, but when they are, they connect different parts of the algebraic landscape. Think of them as flexible pathways that sometimes hit dead ends.

These maps generalize morphisms, allowing for more freedom in how varieties relate. They're defined by rational functions, which are like algebraic fractions. Understanding rational maps is key to navigating the twists and turns of algebraic geometry.

Rational Maps Between Varieties

Definition and Domain

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  • A between two algebraic varieties XX and YY is a map defined by rational functions on XX
    • Rational functions are quotients of polynomial functions on XX
  • The of definition of a rational map is an open dense subset UU of XX where all the defining rational functions are regular (well-defined)
  • A rational map f:X⇢Yf: X ⇢ Y is denoted by a dotted arrow to emphasize that it may not be defined everywhere on XX
  • The graph of a rational map f:X⇢Yf: X ⇢ Y is the closure of the set {(x,f(x))∣x∈U}\{(x, f(x)) | x ∈ U\} in the product variety X×YX × Y

Generalization of Morphisms

  • Rational maps are a generalization of morphisms between varieties
    • They allow for maps that are not necessarily everywhere defined
  • Every of varieties is a rational map, but not every rational map is a morphism
  • A rational map f:X⇢Yf: X ⇢ Y is a morphism if and only if it is defined everywhere on XX, i.e., its domain of definition is the entire variety XX

Composition of Rational Maps

Definition and Existence

  • The composition of two rational maps f:X⇢Yf: X ⇢ Y and g:Y⇢Zg: Y ⇢ Z, denoted by g∘f:X⇢Zg ∘ f: X ⇢ Z, is defined by composing the corresponding rational functions
  • For the composition g∘fg ∘ f to exist, the image of the domain of definition of ff should not be entirely contained in the locus where gg is undefined
  • The domain of definition of the composition g∘fg ∘ f is the preimage under ff of the domain of definition of gg, intersected with the domain of definition of ff

Composition with Complete Varieties

  • If f:X⇢Yf: X ⇢ Y and g:Y⇢Zg: Y ⇢ Z are rational maps, and YY is a complete variety, then the composition g∘f:X⇢Zg ∘ f: X ⇢ Z always exists as a rational map
    • Complete varieties include projective varieties and proper varieties
  • The completeness of the intermediate variety YY ensures that the composition is always well-defined as a rational map

Rational Maps vs Morphisms

Relationship and Differences

  • A morphism of varieties f:X→Yf: X → Y is a continuous map that is locally given by polynomial functions
  • The set of all morphisms between two varieties XX and YY forms a subset of the set of all rational maps between XX and YY
  • A rational map that is defined everywhere on its domain is a morphism, while a morphism is always a rational map

Examples

  • The projection map from a projective space to a lower-dimensional projective space is a morphism and a rational map
  • A birational map between two varieties (a rational map with a rational inverse) is not necessarily a morphism, as it may not be defined everywhere

Behavior of Rational Maps at Undefined Points

Indeterminacy Locus

  • The locus where a rational map f:X⇢Yf: X ⇢ Y is not defined is called the indeterminacy locus of ff, denoted by I(f)I(f)
  • The indeterminacy locus I(f)I(f) is a closed subvariety of XX of codimension at least 2
  • A rational map f:X⇢Yf: X ⇢ Y can be extended to a morphism from an open subset UU of XX to YY if and only if the indeterminacy locus I(f)I(f) has codimension at least 2 in XX

Resolving Indeterminacy by Blowups

  • The behavior of a rational map near a point in its indeterminacy locus can be studied by blowing up the variety XX at that point
    • Blowing up a variety replaces a point with a projective space of directions emanating from that point
  • The induced rational map on the blowup can provide insights into the geometry of the map and the varieties involved
  • Resolving the indeterminacy of a rational map by blowing up the variety can help understand the map's behavior and extend it to a larger domain

Key Terms to Review (17)

Affine variety: An affine variety is a subset of affine space that is defined as the common zero set of a collection of polynomials. It represents the solution set to polynomial equations, allowing for the study of geometric properties using algebraic techniques, and serves as a fundamental building block in algebraic geometry.
Birational equivalence: Birational equivalence is a relationship between algebraic varieties where two varieties are considered equivalent if they can be connected by rational maps that are inverses of each other on dense open subsets. This concept is fundamental in understanding how varieties can share similar properties, even if they are not isomorphic as schemes. Birational equivalence often arises in the study of projective varieties and their properties, where it helps in classifying varieties based on their geometric features.
Blow-up: A blow-up is a fundamental operation in algebraic geometry that allows us to resolve singularities of a variety by replacing it with a new variety. This process transforms a point on the original variety into a higher-dimensional space, effectively 'spreading out' the structure around that point, which can lead to improved properties like smoothness. Blow-ups are closely connected to birational equivalence and rational maps as they provide a way to create new varieties that can often be related through these concepts.
David Mumford: David Mumford is a renowned mathematician recognized for his significant contributions to algebraic geometry, particularly in intersection theory and the study of moduli spaces. His work laid the foundation for modern approaches to intersection multiplicity, birational geometry, and the understanding of rational maps between varieties.
Domain: In algebraic geometry, the domain refers to the set of points in a variety where a rational map is defined and behaves well. It captures the concept of where you can evaluate the rational function without encountering undefined behavior, such as division by zero or discontinuities. Understanding the domain is crucial when analyzing rational maps between varieties, as it helps identify valid inputs for the functions involved.
Francesco Severi: Francesco Severi was an influential Italian mathematician known for his contributions to algebraic geometry and the theory of rational maps between varieties. His work laid foundational concepts that helped bridge various areas in mathematics, particularly focusing on how algebraic structures can be understood through geometric perspectives.
Function Field: A function field is a field that consists of functions defined on a variety, similar to how rational numbers are a field of numbers. These fields provide a framework for studying geometric properties of varieties by allowing one to perform algebraic operations on functions. In algebraic geometry, function fields are essential in understanding rational maps between varieties and also play a key role in establishing the connections made in foundational theorems.
Indeterminate Points: Indeterminate points are specific points in the context of rational maps between varieties where the map is not well-defined, often resulting from a lack of correspondence due to the map's algebraic structure. These points arise when the denominator of a rational function equals zero, leading to undefined behavior in the mapping from one variety to another. Understanding indeterminate points is crucial for analyzing the properties and behaviors of rational maps in algebraic geometry.
Morphism: A morphism is a structure-preserving map between two algebraic objects, such as varieties or algebraic sets, that allows us to understand their relationship in a geometric and algebraic context. Morphisms play a crucial role in linking different varieties and understanding their properties, enabling us to study their intersections, projections, and other geometric features.
Projection from a line: Projection from a line refers to a geometric operation where points in a space are mapped onto a specified line, effectively translating their positions while maintaining a consistent relationship with the line. This operation is important when analyzing the relationships between geometric objects and understanding rational maps, as it provides a way to simplify complex structures by reducing dimensions.
Projective Variety: A projective variety is a subset of projective space that can be defined as the common zeros of homogeneous polynomials. These varieties have a rich structure, enabling the study of geometric properties that can be translated into algebraic terms, making them central to various advanced concepts in algebraic geometry.
Pullback: A pullback is a mathematical operation that takes a function defined on one space and 'pulls it back' to another space via a map between those spaces. This concept is essential for understanding how functions interact with different structures, especially in the context of sheaves and rational maps, allowing for a way to transport or relate local data from one space to another.
Pushforward: The pushforward is a fundamental operation in algebraic geometry that allows one to transfer geometric and algebraic data from one space to another via a morphism or a map. It enables the transformation of sheaves, functions, and other structures along a given map, capturing how properties and relationships are preserved or altered between different varieties. Understanding pushforward helps in analyzing intersections, rational maps, and their impact on the structures involved.
Rational map: A rational map is a function between varieties that is defined by polynomials but may not be well-defined everywhere, meaning it can have undefined points. This concept connects various areas of algebraic geometry, as rational maps can be used to study relationships between varieties, allowing the examination of their properties, singularities, and geometric structures. They play a crucial role in understanding projective varieties, as well as in techniques like blowing up to resolve singularities.
Regular function: A regular function is a type of function defined on a variety that behaves nicely, meaning it is represented by a polynomial and does not have any singularities. These functions are crucial in algebraic geometry because they allow for a well-defined structure on the varieties, facilitating the study of their properties. Regular functions can be thought of as the 'nice' functions that can be computed at every point of a variety, offering important insights into the geometric properties of the space.
Target: In the context of rational maps between varieties, a target refers to the codomain or image of a rational map, which connects two algebraic varieties. The target is crucial in understanding how points from the domain (the source variety) map to points in the target variety, revealing the structure and properties of both varieties involved. Rational maps help identify how one variety can be represented or transformed in relation to another.
Zariski's Main Theorem: Zariski's Main Theorem establishes a deep connection between the geometry of algebraic varieties and the structure of their function fields. It states that for a proper morphism between varieties, the field of rational functions on the source variety can be related to the field of rational functions on the target variety through the process of pullback, ensuring that certain properties, like birationality, are preserved. This theorem is foundational in understanding how rational maps behave in algebraic geometry, particularly during blow-ups and resolutions of singularities.
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