5.3 Rational maps and birational equivalence

Rational maps and birational equivalence are key concepts in algebraic geometry. They allow us to study relationships between varieties that aren't necessarily isomorphic. These ideas help us understand the structure of varieties by focusing on their function fields rather than their specific geometric properties.

Birational equivalence is like a looser version of isomorphism. It lets us relate varieties that are "almost the same" in some sense. This concept is crucial for classification problems and for understanding the intrinsic properties of varieties that don't depend on specific embeddings or coordinate systems.

Rational Maps Between Varieties

Definition and Local Representation

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• A rational map between two varieties $X$ and $Y$ is a morphism defined on an open dense subset $U$ of $X$, taking values in $Y$
• Rational maps are locally represented by quotients of regular functions
  • For affine varieties, a rational map is given by a tuple of rational functions $(f_1/g_1, \ldots, f_n/g_n)$ where $f_i, g_i$ are regular functions on $X$ and $g_i$ are not simultaneously zero on $U$
  • For projective varieties, a rational map is given by a tuple of homogeneous polynomials $(F_0: \ldots: F_n)$ of the same degree, not all simultaneously zero on $U$

Domain of Definition and Indeterminacy Locus

• The domain of definition of a rational map is the largest open subset where the map is defined
  • It is the complement of the set of points where all the defining regular functions (or homogeneous polynomials) vanish simultaneously
• The indeterminacy locus of a rational map is the complement of its domain of definition, consisting of points where the map is not defined
  • For a rational map given by $(f_1/g_1, \ldots, f_n/g_n)$, the indeterminacy locus is the set of points where all $g_i$ vanish simultaneously
  • The indeterminacy locus is a closed subset of $X$ of codimension at least 2

Birational Equivalence and Maps

Definition and Equivalence Relation

• Two varieties $X$ and $Y$ are birationally equivalent if there exist rational maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f$ and $f \circ g$ are the identity maps on dense open subsets of $X$ and $Y$, respectively
  • In other words, $f$ and $g$ are inverse to each other on dense open subsets
• A birational map is a rational map that admits an inverse rational map, establishing a birational equivalence between the varieties
• Birational equivalence is an equivalence relation on the class of varieties
  • Reflexivity: Every variety is birationally equivalent to itself via the identity map
  • Symmetry: If $X$ is birationally equivalent to $Y$, then $Y$ is birationally equivalent to $X$ by swapping the roles of the rational maps
  • Transitivity: If $X$ is birationally equivalent to $Y$ and $Y$ is birationally equivalent to $Z$, then $X$ is birationally equivalent to $Z$ by composing the corresponding rational maps

Function Field Isomorphism

• Birational maps preserve the function field of varieties, inducing an isomorphism between the function fields
  • If $f: X \rightarrow Y$ is a birational map, then it induces an isomorphism $f^*: K(Y) \rightarrow K(X)$ between the function fields
  • The induced isomorphism preserves the field operations and the transcendence degree over the base field
• Two varieties are birationally equivalent if and only if their function fields are isomorphic
  • This provides an algebraic characterization of birational equivalence

Proving Birational Equivalence

Constructing Explicit Rational Maps

• To prove that two varieties $X$ and $Y$ are birationally equivalent, one needs to construct explicit rational maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ and show that their compositions are the identity maps on dense open subsets
  • This involves finding suitable expressions for the rational maps in terms of the coordinates or generators of the function fields
  • The maps should be defined on dense open subsets and satisfy the composition property $g \circ f = \text{id}_X$ and $f \circ g = \text{id}_Y$ on these subsets

Dimension and Function Field Isomorphism

• The dimension of birationally equivalent varieties must be the same
  • Birational maps preserve the transcendence degree of the function field over the base field, which equals the dimension of the variety
• Birational equivalence can be established by finding an isomorphism between the function fields of the varieties
  • If an isomorphism $\varphi: K(X) \rightarrow K(Y)$ is found, it induces birational maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ that realize the birational equivalence
  • The maps $f$ and $g$ can be obtained by expressing the generators of one function field in terms of the generators of the other

Properties Not Preserved

• Certain geometric properties, such as smoothness or projectivity, may not be preserved under birational equivalence
  • Example: The affine line $\mathbb{A}^1$ and the punctured affine plane $\mathbb{A}^2 \setminus \{(0, 0)\}$ are birationally equivalent, but the former is smooth while the latter is not
  • Example: The projective line $\mathbb{P}^1$ and the affine line $\mathbb{A}^1$ are birationally equivalent, but the former is projective while the latter is not

Properties of Rational Maps

Degree and Preimage Count

• The degree of a rational map is the degree of the corresponding function field extension
  • If $f: X \rightarrow Y$ is a rational map, the degree of $f$ is the degree of the field extension $[K(X): f^* K(Y)]$
• For a rational map $f: X \rightarrow Y$ between projective varieties, the degree can be computed as the number of preimages of a general point in $Y$, counted with multiplicity
  • A general point means a point outside a certain closed subset of $Y$ of codimension at least 2
  • The preimages are counted with multiplicity according to the order of vanishing of the defining homogeneous polynomials at each point

Base Points and Resolution

• Base points of a rational map are points in the domain where all the defining polynomials vanish simultaneously
  • For a rational map given by $(F_0: \ldots : F_n)$, the base points are the common zeros of all $F_i$
• Blowing up the base points of a rational map can resolve the indeterminacy and yield a morphism
  • The blow-up replaces each base point with a projective space of dimension one less than the variety
  • The blow-up map is a birational morphism that resolves the indeterminacy of the rational map
  • The resulting morphism is called a resolution of the rational map

Composition and Degree Multiplicativity

• The behavior of rational maps under composition, such as the multiplicativity of degrees, can be studied
  • If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are rational maps, then the composition $g \circ f: X \rightarrow Z$ is also a rational map
  • The degree of the composition is the product of the degrees of the individual maps: $\deg(g \circ f) = \deg(g) \cdot \deg(f)$
  • This property follows from the multiplicativity of degrees in field extensions