To give a rational function f in the function field K(X) of a smooth projective curve X is equivalent to giving a finite morphism f:X→PP1. What is the precise analogue of this statement for higher-dimensional varieties? I would guess that a rational function on a smooth projective variety gives rise to a rational morphism X→PP1 and that this is a morphism up to blowing-up X. Is that correct?

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To give a rational function f in the function field K(X) of a smooth projective curve X is equivalent to giving a finite morphism f:X→PP1.    What is the precise analogue of this statement for higher-dimensional varieties?    I would guess that a rational function on a smooth projective variety gives rise to a rational morphism X→PP1 and that this is a morphism up to blowing-up X. Is that correct?

Yosef Krause · Accepted Answer

If X is an integral scheme of finite type over a field k (k-variety), there is a bijection    K(X)=∼{Rational functions X→PP1}.    The functor K(-) establishes a duality between the category of    k-varieties, with dominant rational maps,    and the category of    function fields over k.    In particular, to a dominant rational map f:X→PP1 there corresponds a monomorphism f⋅:k(t)→K(X). The image of t under f⋅ &quot;is&quot; the rational function f.

To give a rational function f in the function field K(X) of a smooth projective curve X is equivalen

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