= Finite normalization model of a one-variable function field
{title2=$K/k,\quad\operatorname{trdeg}_kK=1$}
For an <algebraically closed field> $k$ and a finitely generated <field extension> $K/k$ of <transcendence degree> one, choose $t$ with $K/k(t)$ finite. The <integral closures> of the two standard coordinate rings of the <projective line> in $K$ are finite modules and glue to a finite <normalization of an algebraic curve> of the projective line. A one-dimensional normal local ring is a <discrete valuation ring>; over the perfect field $k$ the resulting projective curve is smooth. Extension of rational maps gives uniqueness of this smooth projective model.
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