4.4 Function fields and rational functions

Function fields and rational functions are the backbone of algebraic geometry. They're like the DNA of algebraic varieties, containing all the essential info about their structure and properties. Understanding these concepts is key to unlocking the secrets of geometric objects.

In this part of the chapter, we dive into the nitty-gritty of function fields and rational functions. We'll explore their definitions, properties, and how they relate to birational equivalence. Plus, we'll see how these ideas apply to real-world problems in algebraic geometry.

Function fields of varieties

Definition and elements of function fields

• The function field of an algebraic variety $X$, denoted $K(X)$, is the field of rational functions on $X$
• Elements of the function field $K(X)$ are equivalence classes of pairs $(U, f)$, where:
  • $U$ is a non-empty open subset of $X$
  • $f$ is a regular function on $U$
• Two pairs $(U, f)$ and $(V, g)$ are equivalent if $f$ and $g$ agree on the intersection of $U$ and $V$
• The function field $K(X)$ is a field extension of the base field $k$, and its elements are called rational functions on $X$
• The function field $K(X)$ is the smallest field containing all regular functions on $X$

Properties and examples of function fields

• The function field $K(X)$ contains the field of regular functions on $X$ as a subfield
• For an affine variety $X$ defined by the ideal $I(X)$ in the polynomial ring $k[x_1, \ldots, x_n]$, the function field $K(X)$ is isomorphic to the field of fractions of the coordinate ring $k[x_1, \ldots, x_n]/I(X)$
• Example: For the affine plane curve $X$ defined by the equation $y^2 = x^3 - x$, the function field $K(X)$ is isomorphic to $k(t)$, the field of rational functions in one variable $t$, via the isomorphism sending $x$ to $t^2$ and $y$ to $t(t^2 - 1)$
• The function field of a projective variety can be defined similarly using homogeneous coordinates
• Example: For the projective plane curve $X$ defined by the homogeneous equation $x^3 + y^3 = z^3$, the function field $K(X)$ is isomorphic to $k(s, t)$, the field of rational functions in two variables $s$ and $t$, via the isomorphism sending $x/z$ to $s$ and $y/z$ to $t$

Rational functions and their properties

Definition and representation of rational functions

• A rational function on an algebraic variety $X$ is an element of the function field $K(X)$
• Rational functions can be represented as the ratio of two polynomial functions, i.e., $f/g$, where:
  • $f$ and $g$ are regular functions on $X$
  • $g$ is not identically zero
• The set of rational functions on $X$ forms a field under the usual operations of addition and multiplication of functions

Domains, poles, and examples of rational functions

• Rational functions are not necessarily defined everywhere on $X$, as they may have poles (points where the denominator vanishes)
• The domain of a rational function is the largest open subset of $X$ where the function is defined
• Example: On the affine plane $\mathbb{A}^2$, the rational function $f(x, y) = (x^2 + y^2)/(x - 1)$ has a pole at the point $(1, y)$ for any $y$, and its domain is $\mathbb{A}^2 \setminus \{(1, y) : y \in k\}$
• Rational functions can be used to define maps between algebraic varieties
• Example: The rational function $f(x, y) = (x, y/x)$ defines a rational map from the affine plane $\mathbb{A}^2$ to itself, which is not defined at the points $(0, y)$ for any $y$

Function fields and birational equivalence

Birational equivalence and isomorphisms of function fields

• Two algebraic varieties $X$ and $Y$ are birationally equivalent if there exist rational maps $f: X \to Y$ and $g: Y \to X$ such that:
  • $g \circ f$ and $f \circ g$ are the identity maps on dense open subsets of $X$ and $Y$, respectively
• If $X$ and $Y$ are birationally equivalent, then their function fields $K(X)$ and $K(Y)$ are isomorphic as fields
• The converse is also true: if the function fields $K(X)$ and $K(Y)$ are isomorphic, then $X$ and $Y$ are birationally equivalent

Properties and examples of birational equivalence

• Birational equivalence is an equivalence relation on the set of algebraic varieties, and it preserves many geometric properties of the varieties
• Example: The affine plane curve $X$ defined by $y^2 = x^3 - x$ and the affine line $\mathbb{A}^1$ are birationally equivalent via the rational maps $f: X \to \mathbb{A}^1$, $(x, y) \mapsto x$, and $g: \mathbb{A}^1 \to X$, $t \mapsto (t^2, t(t^2 - 1))$
• The study of birational equivalence leads to the concept of birational geometry, which focuses on properties that are invariant under birational maps
• Example: The birational geometry of algebraic surfaces is closely related to the minimal model program, which aims to classify surfaces up to birational equivalence by finding their minimal models (smooth surfaces with nef canonical divisor)

Applications of function fields in algebraic geometry

Algebraic curves and the Riemann-Roch theorem

• Function fields and rational functions are essential tools in the study of algebraic curves and surfaces
• The genus of an algebraic curve can be determined using the Riemann-Roch theorem, which relates the dimension of the space of rational functions with prescribed poles to the genus of the curve
• Example: For a smooth projective curve $X$ of genus $g$, the Riemann-Roch theorem states that $\dim L(D) - \dim L(K - D) = \deg D - g + 1$, where $L(D)$ is the space of rational functions with poles bounded by the divisor $D$, and $K$ is the canonical divisor of $X$

Canonical divisors, singularities, and birational transformations

• The canonical divisor of an algebraic curve or surface can be defined using rational functions, and it plays a crucial role in the classification of algebraic varieties
• Rational functions can be used to study the singularities of algebraic varieties and to resolve them through birational transformations (e.g., blow-ups)
• Example: The blow-up of the affine plane $\mathbb{A}^2$ at the origin is a birational transformation that resolves the singularity of the curve $y^2 = x^3$ at $(0, 0)$, resulting in a smooth curve in the blown-up surface

Algebraic cycles, intersection theory, and cohomology

• The theory of function fields and rational functions is applied in the study of algebraic cycles, intersection theory, and the computation of cohomology groups of algebraic varieties
• Example: The Chow ring of a smooth projective variety $X$ is a graded ring whose elements are algebraic cycles modulo rational equivalence, and the intersection product of cycles can be defined using rational functions
• The cohomology groups of an algebraic variety can be computed using the Čech cohomology of a cover by affine open subsets, which involves the function fields of the open subsets
• Example: The de Rham cohomology of a smooth affine variety $X$ can be computed using the complex of regular differential forms on $X$, which are rational functions with certain properties