1.3 Morphisms and rational maps

Morphisms and rational maps are the backbone of algebraic geometry, connecting varieties and unlocking their secrets. They're like the highways and backroads of math, letting us travel between different geometric landscapes and explore their hidden connections.

Regular maps are the smooth highways, defined everywhere with polynomial precision. Rational maps are the adventurous backroads, allowing for more flexibility but sometimes hitting dead ends. Together, they give us powerful tools to navigate the intricate world of algebraic varieties.

Regular vs Rational Maps

Definitions and Distinctions

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• A regular map (or morphism) between affine varieties is a map that can be defined by polynomials in each coordinate
  • For example, the map $f: \mathbb{A}^1 \to \mathbb{A}^2$ given by $f(t) = (t^2, t^3)$ is a regular map
• A rational map between affine varieties is a map defined by rational functions (quotients of polynomials) in each coordinate, with the condition that the denominator does not vanish on some open dense subset of the domain
  • For example, the map $g: \mathbb{A}^1 \to \mathbb{A}^1$ given by $g(t) = \frac{t^2 - 1}{t + 1}$ is a rational map, defined on the open dense subset $\{t \neq -1\}$
• For projective varieties, a regular map is defined by homogeneous polynomials of the same degree in each coordinate, while a rational map is defined by homogeneous rational functions of the same degree
  • For example, the map $h: \mathbb{P}^1 \to \mathbb{P}^2$ given by $h([s:t]) = [s^2:st:t^2]$ is a regular map, while the map $k: \mathbb{P}^1 \to \mathbb{P}^1$ given by $k([s:t]) = [s^2-t^2:st]$ is a rational map

Properties and Extensions

• The set of regular maps between two varieties forms a ring, called the ring of regular functions
  • This ring captures the algebraic structure of the varieties and their morphisms
• Rational maps can be extended to regular maps by considering the graph of the rational map and taking its closure in the product of the domain and codomain varieties
  • This process allows us to study rational maps using the tools developed for regular maps
  • For example, the rational map $g$ above can be extended to a regular map $\overline{g}: \mathbb{A}^1 \to \mathbb{P}^1$ given by $\overline{g}(t) = [t^2-1:t+1]$, where $[-1:1]$ is the point at infinity corresponding to the direction of approach to $t=-1$

Composition and Invertibility of Morphisms

Categorical Structure

• The composition of two regular maps (or morphisms) is again a regular map, making the set of varieties into a category with morphisms as the arrows
  • This categorical structure allows us to study varieties and their relationships using the powerful tools of category theory
  • For example, if $f: X \to Y$ and $g: Y \to Z$ are regular maps, then their composition $g \circ f: X \to Z$ is also a regular map
• A morphism is an isomorphism if it has an inverse morphism, i.e., there exists another morphism such that their composition (in both orders) is the identity morphism
  • Isomorphisms capture the notion of algebraic equivalence between varieties
  • For example, the map $f: \mathbb{A}^1 \to \mathbb{A}^1$ given by $f(t) = t^3$ is an isomorphism, with inverse $f^{-1}(u) = \sqrt[3]{u}$

Automorphisms and Symmetries

• A morphism is an automorphism if it is an isomorphism from a variety to itself
  • Automorphisms capture the symmetries and self-equivalences of a variety
  • For example, the map $f: \mathbb{A}^1 \to \mathbb{A}^1$ given by $f(t) = -t$ is an automorphism of the affine line
• The set of automorphisms of a variety forms a group under composition, called the automorphism group of the variety
  • This group encodes the symmetries and self-transformations of the variety
  • For example, the automorphism group of the projective line $\mathbb{P}^1$ is the projective linear group $PGL(2, \mathbb{C})$, consisting of fractional linear transformations $t \mapsto \frac{at+b}{ct+d}$ with $ad-bc \neq 0$

Image and Preimage of a Morphism

Definitions and Properties

• The image of a morphism is the set of all points in the codomain that are mapped to by some point in the domain
  • For example, the image of the map $f: \mathbb{A}^1 \to \mathbb{A}^2$ given by $f(t) = (t^2, t^3)$ is the curve $\{(x, y) \in \mathbb{A}^2 \mid y^2 = x^3\}$
• The preimage (or inverse image) of a subset of the codomain under a morphism is the set of all points in the domain that map to a point in that subset
  • For example, the preimage of the point $(1, 1)$ under the map $f$ above is the set $\{1, -1\}$
• The image of a morphism is always a constructible set, i.e., a finite union of locally closed sets (intersections of open and closed sets)
  • This property allows us to study the image using the tools of algebraic geometry and topology
• The preimage of a closed set under a morphism is always closed, and the preimage of an open set is always open (this is the definition of a continuous map in the Zariski topology)
  • This property ensures that morphisms behave well with respect to the Zariski topology on varieties

Applications and Examples

• The image and preimage of a morphism can be used to study the fibers of the morphism, i.e., the preimages of individual points in the codomain
  • For example, the fibers of the map $f: \mathbb{A}^1 \to \mathbb{A}^1$ given by $f(t) = t^2$ are either singletons $\{a\}$ for $a \neq 0$ or the pair $\{0, -0\}$ for $a = 0$
• The image and preimage can also be used to study the ramification and branch locus of a morphism, which capture the points where the morphism fails to be a local isomorphism
  • For example, the map $f: \mathbb{A}^1 \to \mathbb{A}^1$ given by $f(t) = t^2$ is ramified at $t = 0$, and the image point $f(0) = 0$ is a branch point

Birational Equivalence

Definition and Equivalence Relation

• Two varieties are birationally equivalent (or birational) if there exist rational maps between them that are inverses of each other on some open dense subsets
  • This captures the idea that the varieties are "almost isomorphic" or "isomorphic up to codimension 1"
  • For example, the affine plane curve $\{(x, y) \in \mathbb{A}^2 \mid y^2 = x^2(x+1)\}$ is birationally equivalent to the affine line $\mathbb{A}^1$ via the maps $f(x) = (x, x\sqrt{x+1})$ and $g(x, y) = x$, which are inverses on the open dense subsets $\{x \neq 0, -1\}$ and $\{y \neq 0\}$, respectively
• Birational equivalence is an equivalence relation on the set of varieties, and the equivalence classes are called birational equivalence classes or birational types
  • This allows us to classify varieties up to birational equivalence and study their birational geometry

Function Fields and Birational Maps

• Birational varieties have isomorphic function fields, i.e., the fields of rational functions on the varieties are isomorphic
  • This provides an algebraic characterization of birational equivalence
  • For example, the function field of the plane curve above is isomorphic to the field of rational functions $\mathbb{C}(x)$ on the affine line
• A rational map that is an isomorphism between open dense subsets of two varieties is called a birational map or a birational isomorphism
  • Birational maps allow us to identify varieties up to birational equivalence and transfer properties between them
  • For example, the map $f(x, y) = (x, xy)$ is a birational isomorphism between the plane curve $\{(x, y) \in \mathbb{A}^2 \mid y^2 = x^2(x+1)\}$ and the affine plane $\mathbb{A}^2$

Resolving Singularities and Birational Models

• The process of finding a variety birational to a given variety but with simpler or more desirable properties (e.g., smoothness, projectivity) is called resolving singularities or finding a birational model
  • This is a fundamental problem in algebraic geometry, with deep connections to classification theory and moduli spaces
  • For example, the plane curve $\{(x, y) \in \mathbb{A}^2 \mid y^2 = x^3\}$ has a singular point at the origin, but it is birationally equivalent to the smooth curve $\{(s, t) \in \mathbb{A}^2 \mid t^2 = s^2(s+1)\}$ via the maps $f(s) = (s^2, s^3)$ and $g(x, y) = (\sqrt{x}, \frac{y}{x})$
• Birational models can be used to study the geometry and invariants of singular varieties by relating them to simpler or better-understood varieties
  • For example, the minimal model program seeks to find a "simplest" birational model for a given variety, with mild singularities and nef canonical divisor