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We know that (for an algebraically closed field $k$) there is an equivalence between algebraic curves over $k$ (up to birational equivalence) and fields of transcendence degree $1$ over $k$.

Is there something similar for higher dimensional algebraic varieties?

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For algebraic varieties the transcendence degree of the function field is equal to the dimension, so the answer is yes: just replace $1$ with $n := \dim X$ – Francesco Polizzi Jun 19 '12 at 16:14
Is this a functorial equivalence? Where can I read about this? – expmat Jun 19 '12 at 16:15
Hartshorne, Chapter I, Section 4 – Francesco Polizzi Jun 19 '12 at 16:21
In your "we know that" sentence, you really need the fields of transcendence degree 1 to be finitely generated. – S. Carnahan Jun 20 '12 at 8:30

For $X,Y$ integral $k$-varieties, there is a bijection {$f: X \to Y$ dominant rational} <-> $\mathrm{Hom}_k(K(Y),K(X))$.

Also, there is an equivalence of categories {integral $k$-varieties with morphisms as above} and {finitely generated field extensions of $k$}.

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