algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, the smooth completion (or smooth compactification) of a smooth affine algebraic curve ''X'' is a

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

smooth

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

which contains ''X'' as an open subset. Smooth completions exist and are unique over a

perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has no multiple roots in any field extension ''F/k''. * Every irreducible polynomial over ''k'' has non-zero f ...

Examples

An affine form of a

hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...

may be presented as

y^2=P(x)

where

(x, y)\in\mathbb^2

and () has distinct roots and has degree at least 5. The Zariski closure of the affine curve in

\mathbb\mathbb^2

is singular at the unique infinite point added. Nonetheless, the affine curve can be embedded in a unique compact

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

called its smooth completion. The projection of the Riemann surface to

\mathbb\mathbb^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 In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

of ''X'' is free with

2g+r-1

generators if ''r''>0. (Analogue of

Dirichlet's unit theorem In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a pos ...

) 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 In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the poin ...

the singularities) of the projective curve then gives a smooth completion of ''X''. Their points correspond to the

discrete valuation In mathematics, a discrete valuation is an integer valuation on a field ''K''; that is, a function: :\nu:K\to\mathbb Z\cup\ satisfying the conditions: :\nu(x\cdot y)=\nu(x)+\nu(y) :\nu(x+y)\geq\min\big\ :\nu(x)=\infty\iff x=0 for all x,y\in K ...

s 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 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

Bibliography