(Minimal model theorem for curves)
Let \(k\) be an algebraically closed field. The following categories are equivalent:
smooth projective curves over \(k\) with dominant morphisms;
quasiprojective curves over \(k\) with dominant rational maps;
finitely generated field extensions \(K/k\) of transcendence degree \(1\), with \(k\)-algebra homomorphisms, with arrows reversed.
In particular every curve is birational to a smooth projective curve, unique up to isomorphism, and two smooth projective curves are birational exactly when they are isomorphic.
The smooth projective model of the function field \(K\) is built from its discrete valuation rings: its closed points are the DVRs of \(K\) containing \(k\), and the normalization of any projective model of \(K\) realizes it. A rational map from a smooth curve to a projective variety extends to a morphism, by the valuative criterion applied at each point, which is why birational smooth projective curves are isomorphic. In dimension \(2\) this uniqueness fails: blowing up a point gives a birational smooth projective surface that is not isomorphic, and minimal models are reached by contracting \((-1)\)-curves.