Smooth completion

In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve X is a complete smooth algebraic curve which contains X as an open subset.^[1] Smooth completions exist and are unique over a perfect field.

Examples

An affine form of a hyperelliptic curve may be presented as $y^2=P(x)$ where $(x, y)\in\mathbb{C}^2$ and $P$ ( $x$ ) has distinct roots and has degree at least 5. The Zariski closure of the affine curve in $\mathbb{C}\mathbb{P}^2$ is singular at the unique infinite point added. Nonetheless, the affine curve can be embedded in a unique compact Riemann surface called its smooth completion. The projection of the Riemann surface to $\mathbb{C}\mathbb{P}^2$ is 2-to-1 over the singular point at infinity if $P(x)$ has even degree, and 1-to-1 (but ramified) otherwise.

This smooth completion can also be obtained as follows. Project the affine curve to the affine line using the x-coordinate. Embed the affine line into the projective line, then take the normalization of the projective line in the function field of the affine curve.

Applications

A smooth connected curve over an algebraically closed field is called hyperbolic if $2g-2+r>0$ 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 $2g+r-1$ generators if r>0.

(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.

Construction

Suppose the base field is perfect. Any affine curve X is isomorphic to an open subset of an integral projective (hence complete) curve. Taking the normalization (or blowing up the singularities) of the projective curve then gives a smooth completion of X. Their points correspond to the discrete valuations of the function field that are trivial on the base field.

By construction, the smooth completion is a projective curve which contains the given curve as an everywhere dense open subset, and the added new points are smooth. Such a (projective) completion always exists and is unique.

If the base field is not perfect, a smooth completion of a smooth affine curve doesn't always exist. But the above process always produces a regular completion if we start with a regular affine curve (smooth varieties are regular, and the converse is true over perfect fields). A regular completion is unique and, by the valuative criterion of properness, any morphism from the affine curve to a complete algebraic variety extends uniquely to the regular completion.

Generalization

If X is a separated algebraic variety, a theorem of Nagata^[2] says that X can be embedded as an open subset of a complete algebraic variety. 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. If X is quasi-projective, the smooth completion can be chosen to be projective.

However, contrary to the one-dimensional case, there is no uniqueness of the smooth completion, nor is it canonical.

References

↑ Griffiths, 1972, p. 286.
↑ http://math.stanford.edu/~conrad/papers/nagatafinal.pdf

Bibliography

Griffiths, Phillip A. (1972). "Function theory of finite order on algebraic varieties. I(A)". J. Differential Geometry 6 (3): 285–306. MR 0325999. Zbl 0269.14003.
Hartshorne, Robin (1977). Algebraic geometry. Graduate Texts in Mathematics 52. New York, Heidelberg: Springer-Verlag. ISBN 0387902449. (see chapter 4).

Topics in algebraic curves

Rational curves

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form

Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm

Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law

Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve

Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem

Singularities	Acnode Crunode Cusp Delta invariant Tacnode

Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

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