8.1 Rational maps between varieties

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

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

Generalization of Morphisms

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

Composition of Rational Maps

Definition and Existence

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

Composition with Complete Varieties

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

Rational Maps vs Morphisms

Relationship and Differences

• A morphism of varieties $f: X → Y$ is a continuous map that is locally given by polynomial functions
• The set of all morphisms between two varieties $X$ and $Y$ forms a subset of the set of all rational maps between $X$ and $Y$
• 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 $f: X ⇢ Y$ is not defined is called the indeterminacy locus of $f$, denoted by $I(f)$
• The indeterminacy locus $I(f)$ is a closed subvariety of $X$ of codimension at least 2
• A rational map $f: X ⇢ Y$ can be extended to a morphism from an open subset $U$ of $X$ to $Y$ if and only if the indeterminacy locus $I(f)$ has codimension at least 2 in $X$

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 $X$ 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