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.

## Contents

* 1 Examples
* 2 Applications
* 3 Construction
* 4 Generalization
* 5 See also
* 6 References
* 7 Bibliography

## Examples

An affine form of a hyperelliptic curve may be presented as [math]\displaystyle{ y^2=P(x) }[/math] where [math]\displaystyle{ (x, y)\in\mathbb{C}^2 }[/math] and P(x) has distinct roots and has degree at least 5. The Zariski closure of the affine curve in [math]\displaystyle{ \mathbb{C}\mathbb{P}^2 }[/math] 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 [math]\displaystyle{ \mathbb{C}\mathbb{P}^1 }[/math] is 2-to-1 over the singular point at infinity if [math]\displaystyle{ P(x) }[/math] 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 [math]\displaystyle{ 2g-2+r\gt 0 }[/math] 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 [math]\displaystyle{ 2g+r-1 }[/math] 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.

## See also

* Hyperelliptic curve
* Bolza surface

## References

1. ↑ Griffiths, 1972, p. 286.
2. ↑ "Deligne's notes on Nagata compactifications". Journal of the Ramanujan Mathematical Society 22 (3): 205–257. 2007. https://math.stanford.edu/~conrad/papers/nagatafinal.pdf.

## Bibliography

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

* v
* t
* e

Topics in algebraic curves

Rational curves|

* Five points determine a conic
* Projective line
* Rational normal curve
* Riemann sphere
* Twisted cubic

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|

* De Franchis theorem
* Faltings's theorem
* Hurwitz's automorphisms theorem
* Hurwitz surface
* Hyperelliptic curve

Plane curves|

* AF+BG theorem
* Bézout's theorem
* Bitangent
* Cayley–Bacharach theorem
* Conic section
* Cramer's paradox
* Cubic plane curve
* Fermat curve
* Genus–degree formula
* Hilbert's sixteenth problem
* Nagata's conjecture on curves
* Plücker formula
* Quartic plane curve
* Real plane curve

Riemann surfaces|

* Belyi's theorem
* Bring's curve
* Bolza surface
* Compact Riemann surface
* Dessin d'enfant
* Differential of the first kind
* Klein quartic
* Riemann's existence theorem
* Riemann–Roch theorem
* Teichmüller space
* Torelli theorem

Constructions|

* Dual curve
* Polar curve
* Smooth completion

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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