# Smooth completion > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Smooth_completion > Markdown URL: https://mediated.wiki/source/Smooth_completion.md > Source: https://en.wikipedia.org/wiki/Smooth_completion > Source revision: 1086275533 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [algebraic geometry](/source/algebraic_geometry), the '''smooth completion''' (or '''smooth compactification''') of a [smooth](/source/smooth_scheme) [affine algebraic curve](/source/affine_algebraic_curve) ''X'' is a [complete](/source/complete_variety) smooth [algebraic curve](/source/algebraic_curve) which contains ''X'' as an open subset.Griffiths, 1972, p. 286. Smooth completions exist and are unique over a [perfect field](/source/perfect_field). ==Examples== An affine form of a [hyperelliptic curve](/source/hyperelliptic_curve) may be presented as y^2=P(x) where (x, y)\in\mathbb{C}^2 and {{mvar|P}}({{mvar|x}}) [has distinct roots](/source/separable_polynomial) 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](/source/point_at_infinity) point added. Nonetheless, the affine curve can be embedded in a unique compact [Riemann surface](/source/Riemann_surface) called its smooth completion. The projection of the Riemann surface to \mathbb{C}\mathbb{P}^1 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](/source/fundamental_group) of ''X'' is free with 2g+r-1 generators if ''r''>0. (Analogue of [Dirichlet's unit theorem](/source/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](/source/blowing_up) the singularities) of the projective curve then gives a smooth completion of ''X''. Their points correspond to the [discrete valuation](/source/discrete_valuation)s of the [function field](/source/function_field_of_an_algebraic_variety) that are trivial on the base field. By construction, the smooth completion is a [projective](/source/projective_variety) 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](/source/Glossary_of_scheme_theory) 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](/source/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](/source/Glossary_of_scheme_theory) algebraic variety, a [theorem of Nagata](/source/Nagata's_compactification_theorem){{cite journal | last = Conrad | first = Brian | author-link = Brian Conrad | issue = 3 | journal = Journal of the Ramanujan Mathematical Society | mr = 2356346 | pages = 205–257 | title = Deligne's notes on Nagata compactifications | url = https://math.stanford.edu/~conrad/papers/nagatafinal.pdf | volume = 22 | year = 2007}} 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](/source/Resolution_of_singularities) ''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](/source/Hyperelliptic_curve) *[Bolza surface](/source/Bolza_surface) ==References== {{Reflist}} ==Bibliography== *{{cite journal |author-link=Phillip Griffiths |last=Griffiths |first=Phillip A. |title=Function theory of finite order on algebraic varieties. I(A) |journal=[Journal of Differential Geometry](/source/Journal_of_Differential_Geometry) |year=1972 |volume=6 |issue=3 |pages=285–306 |mr=0325999 |zbl=0269.14003 }} *{{cite book |author-link=Robin Hartshorne |last=Hartshorne |first=Robin |title=Algebraic geometry |series=[Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics) |volume=52 |publisher=Springer-Verlag |location=New York, Heidelberg |year=1977 |isbn=0387902449 }} (see chapter 4). {{Algebraic curves navbox}} Category:Algebraic geometry Category:Riemann surfaces Category:Algebraic curves Category:Birational geometry --- Adapted from the Wikipedia article [Smooth completion](https://en.wikipedia.org/wiki/Smooth_completion) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Smooth_completion?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.