[1] Smooth completions exist and are unique over a perfect field.
Nonetheless, the affine curve can be embedded in a unique compact Riemann surface called its smooth completion.
A smooth connected curve over an algebraically closed field is called hyperbolic if
where g is the genus of the smooth completion and r is the number of added points.
Over an algebraically closed field of characteristic 0, the fundamental group of X is free with
(Analogue of Dirichlet's unit theorem) Let X be a smooth connected curve over a finite field.
Then the units of the ring of regular functions O(X) on X is a finitely generated abelian group of rank r -1.
Taking the normalization (or blowing up the singularities) of the projective curve then gives a smooth completion of X.
If X is moreover smooth and the base field has characteristic 0, then by Hironaka's theorem X can even be embedded as an open subset of a complete smooth algebraic variety, with boundary a normal crossing divisor.
However, contrary to the one-dimensional case, there is no uniqueness of the smooth completion, nor is it canonical.