8.2 Birational equivalence and isomorphisms

Birational equivalence is a key concept in algebraic geometry, comparing varieties that are "almost isomorphic." It allows for differences in lower-dimensional subsets, making it weaker than isomorphism but still preserving important geometric properties.

Understanding birational equivalence helps us study varieties by relating them to simpler ones. It's crucial for classification problems and gives insights into the structure of algebraic varieties, connecting to function fields and rational maps in meaningful ways.

Birational Equivalence of Varieties

Definition and Implications

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• Two algebraic varieties $X$ and $Y$ are birationally equivalent if there exist rational maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f = id_X$ and $f \circ g = id_Y$, where $id_X$ and $id_Y$ are the identity maps on $X$ and $Y$, respectively
• Birational equivalence captures the idea that two varieties are "almost isomorphic," meaning they are isomorphic except on lower-dimensional subsets (singular points, exceptional curves)
• The dimension of birationally equivalent varieties must be the same
• Birational equivalence is a weaker notion than isomorphism, as it allows for the varieties to differ on subsets of lower dimension

Function Fields and Birational Equivalence

• The function fields of birationally equivalent varieties are isomorphic
• If $X$ and $Y$ are birationally equivalent, then $K(X) \cong K(Y)$, where $K(X)$ and $K(Y)$ are the function fields of $X$ and $Y$, respectively
• The isomorphism between function fields induced by birational equivalence preserves the field operations and the transcendence degree
• Conversely, if the function fields of two varieties are isomorphic, then the varieties are birationally equivalent (Zariski's theorem)

Birational Equivalence as an Equivalence Relation

Reflexivity

• For any algebraic variety $X$, the identity map $id_X: X \rightarrow X$ is a birational map, and $id_X \circ id_X = id_X$, showing that $X$ is birationally equivalent to itself
• The reflexive property holds trivially for any variety, as the identity map is always a birational equivalence

Symmetry

• If $X$ and $Y$ are birationally equivalent with rational maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ satisfying $g \circ f = id_X$ and $f \circ g = id_Y$, then $Y$ and $X$ are also birationally equivalent
• The same maps $f$ and $g$ demonstrate the birational equivalence in the opposite direction
• Symmetry follows from the definition of birational equivalence, as the roles of $X$ and $Y$ can be interchanged

Transitivity

• If $X$ and $Y$ are birationally equivalent with rational maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$, and $Y$ and $Z$ are birationally equivalent with rational maps $h: Y \rightarrow Z$ and $k: Z \rightarrow Y$, then $X$ and $Z$ are birationally equivalent
• The composition of birational maps $h \circ f: X \rightarrow Z$ and $g \circ k: Z \rightarrow X$ demonstrates the birational equivalence between $X$ and $Z$
• Transitivity allows for the "chaining" of birational equivalences to establish equivalence between varieties not directly related by a single pair of birational maps

Constructing Birational Maps

Finding Birational Maps

• To construct a birational map between two algebraic varieties $X$ and $Y$, find 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 $X$ and $Y$, respectively, outside of lower-dimensional subsets
• The maps $f$ and $g$ may not be defined everywhere on the varieties, but they should be defined on open dense subsets
• Techniques for constructing birational maps include using projections (projecting from a point or a subvariety), blowups (resolving singularities or indeterminacies), and blowdowns (contracting subvarieties)

Inverses of Birational Maps

• The inverse of a birational map $f: X \rightarrow Y$ is a rational map $g: Y \rightarrow X$ such that $g \circ f = id_X$ and $f \circ g = id_Y$, where $id_X$ and $id_Y$ are the identity maps on $X$ and $Y$, respectively
• Birational maps and their inverses may not be defined everywhere on the varieties, but they are defined on open dense subsets
• To find the inverse of a birational map, one can solve for the pre-image of a general point under the map and express the result as a rational function (e.g., using coordinate functions)
• The inverse of a birational map is unique up to equality on an open dense subset

Birational Equivalence vs Isomorphism

Comparing the Notions

• Isomorphism is a stronger notion than birational equivalence, as it requires the existence of morphisms (regular maps) $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f = id_X$ and $f \circ g = id_Y$ everywhere on the varieties
• Birationally equivalent varieties may have different local structures, such as singularities or different numbers of irreducible components, while isomorphic varieties have the same local structure
• Isomorphic varieties have the same dimension and degree, while birationally equivalent varieties only need to have the same dimension
• Every isomorphism is a birational equivalence, but not every birational equivalence is an isomorphism

Examples Illustrating the Difference

• The projective line $\mathbb{P}^1$ and the affine line $\mathbb{A}^1$ are birationally equivalent but not isomorphic
  • The map $f: \mathbb{A}^1 \rightarrow \mathbb{P}^1$ given by $x \mapsto [x:1]$ and its inverse $g: \mathbb{P}^1 \rightarrow \mathbb{A}^1$ given by $[x:y] \mapsto x/y$ (for $y \neq 0$) demonstrate the birational equivalence
  • However, there is no isomorphism between $\mathbb{P}^1$ and $\mathbb{A}^1$ because they have different global structures ($\mathbb{P}^1$ is compact, while $\mathbb{A}^1$ is not)
• A singular cubic curve and a non-singular cubic curve are birationally equivalent but not isomorphic
  • The normalization map from the non-singular curve to the singular curve is a birational equivalence
  • The curves are not isomorphic because they have different local structures at the singular point