Complete Algebraic Curve

In algebraic geometry, a complete algebraic curve is an algebraic curve that is complete as an algebraic variety.

A projective curve, a dimension-one projective variety, is a complete curve. A complete curve (over an algebraically closed field) is projective. Because of this, over an algebraically closed field, the terms "projective curve" and "complete curve" are usually used interchangeably. Over a more general base scheme, the distinction still matters.

A curve in $\mathbb^3$ is called an (algebraic) space curve, while a curve in $\mathbb^2$ is called a plane curve. By means of a projection from a point, any smooth projective curve can be embedded into $\mathbb^3$; thus, up to a projection, every (smooth) curve is a space curve. Up to a birational morphism, every (smooth) curve can be embedded into $\mathbb^2$ as a nodal curve.

Riemann's existence theorem says that the category of compact Riemann surfaces is equivalent to that of smooth projective curves over the complex numbers.

Throughout the article, a curve mean a complete curve (but not necessarily smooth).

Abstract complete curve

Let ''k'' be an algebrically closed field. By a function field ''K'' over ''k'', we mean a finitely generated field extension of ''k'' that is typically not algebraic (i.e., a transcendental extension). The function field of an algebraic variety is a basic example. For a function field of transcendence degree one, the converse holds by the following construction. Let $C_K$ denote the set of all discrete valuation rings of $K/k$. We put the topology on $C_K$ so that the closed subsets are either finite subsets or the whole space. We then make it a locally ringed space by taking $\mathcal(U)$ to be the intersection $\cap_ R$. Then the $C_K$ for various function fields ''K'' of transcendence degree one form a category that is equivalent to the category of smooth projective curves.

One consequence of the above construction is that a complete smooth curve is projective (since a complete smooth curve of ''C'' corresponds to $C_K, K = k(C)$, which corresponds to a projective smooth curve.)

Smooth completion of an affine curve

Let $C_0 = V(f) \subset \mathbb^2$ be a smooth affine curve given by a polynomial ''f'' in two variables. The closure $\overline$ in $\mathbb^2$, the projective completion of it, may or may not be smooth. The normalization ''C'' of $\overline$ is smooth and contains $C_0$ as an open dense subset. Then the curve $C$ is called the smooth completion of $C_0$. (Note the smooth completion of $C_0$ is unique up to isomorphism since two smooth curves are isomorphic if they are birational to each other.)

For example, if $f = y^2 - x^3 + 1$, then $\overline$ is given by $y^2 z = x^3 - z^3$, which is smooth (by a Jacobian computation). On the other hand, consider $f = y^2 - x^6 + 1$. Then, by a Jacobian computation, $\overline$ is not smooth. In fact, $C_0$ is an (affine) hyperelliptic curve and a hyperelliptic curve is not a plane curve (since a hyperelliptic curve is never a complete intersection in a projective space).

Over the complex numbers, ''C'' is a compact Riemann surface that is classically called the Riemann surface associated to the algebraic function $y(x)$ when $f(x, y(x)) \equiv 0$. Conversely, each compact Riemann surface is of that form; this is known as the Riemann existence theorem.

A map from a curve to a projective space

To give a rational map from a (projective) curve ''C'' to a projective space is to give a linear system of divisors ''V'' on ''C'', up to the fixed part of the system? (need to be clarified); namely, when ''B'' is the base locus (the common zero sets of the nonzero sections in ''V''), there is:
:$f: C - B \to \mathbb(V^*)$
that maps each point $P$ in $C - B$ to the hyperplane $\$. Conversely, given a rational map ''f'' from ''C'' to a projective space,

In particular, one can take the linear system to be the canonical linear system $, K, = \mathbb(\Gamma(C, \omega_C))$ and the corresponding map is called the canonical map.

Let $g$ be the genus of a smooth curve ''C''. If $g = 0$, then $, K,$ is empty while if $g = 1$, then $, K, = 0$. If $g \ge 2$, then the canonical linear system $, K,$ can be shown to have no base point and thus determines the morphism $f : C \to \mathbb^$. If the degree of ''f'' or equivalently the degree of the linear system is 2, then ''C'' is called a hyperelliptic curve.

Max Noether's theorem implies that a non-hyperelliptic curve is projectively normal when it is embedded into a projective space by the canonical divisor.

Classification of smooth algebraic curves in $\mathbb^3$

The classification of a smooth projective curve begins with specifying a genus. For genus zero, there is only one: the projective line $\mathbb^1$ (up to isomorphism). A genus-one curve is precisely an elliptic curve and isomorphism classes of elliptic curves are specified by a j-invariant (which is an element of the base field). The classification of genus-2 curves is much more complicated; here is some partial result over an algebraically closed field of characteristic not two:
*Each genus-two curve ''X'' comes with the map $f: X \to \mathbb^1$ determined by the canonical divisor; called the canonical map. The canonical map has exactly 6 ramified points of index 2.
*Conversely, given distinct 6 points $a_1, \dots, a_6$, let $K$ be the field extension of $k(x)$, ''x'' a variable, given by the equation $y^2 = (x-a_1) \cdots (x-a_6)$ and $f : X \to \mathbb^1$ the map corresponding to the extension. Then $X$ is a genus-two curve and $f$ ramifies exactly over those six points.

For genus $\ge 3$, the following terminology is used:
*Given a smooth curve ''C'', a divisor ''D'' on it and a vector subspace $V \subset H^0(C, \mathcal(D))$, one says the linear system $\mathbb(V)$ is a g^r_d if ''V'' has dimension ''r''+1 and ''D'' has degree ''d''. One says ''C'' has a g^r_d if there is such a linear system.

Fundamental group

Let ''X'' be a smooth complete algebraic curve. Then the étale fundamental group of ''X'' is defined as:
:$\pi_1(X) = \varprojlim_ \operatorname(L/K)$
where $K$ is the function field of ''X'' and $L/K$ is a Galois extension.

Specific curves

Canonical curve

If ''X'' is a nonhyperelliptic curve of genus $\ge 3$, then the linear system $, K,$ associated to the canonical divisor is very ample; i.e., it gives an embedding into the projective space. The image of that embedding is then called a canonical curve.

Stable curve

A stable curve is a connected nodal curve with finite automorphism group.

Spectral curve

Vector bundles on a curve

Line bundles and dual graph

Let ''X'' be a possibly singular curve over complex numbers. Then
:$0 \to \mathbb^* \to (\mathbb^*)^r \to \Gamma(X, \mathcal) \to \operatorname(X) \to \operatorname(\widetilde) \to 0.$
where ''r'' is the number of irreducible components of ''X'', $\pi:\widetilde \to X$ is the normal scheme, normalization and $\mathcal = \pi_* \mathcal_/\mathcal_X$. (To get this use the fact $\operatorname(X) = \operatorname^1(X, \mathcal_X^*)$ and $\operatorname(\widetilde) = \operatorname^1(\widetilde, \mathcal_^*) = \operatorname^1(X, \pi_* \mathcal_^*).$)

Taking the long exact sequence of the exponential sheaf sequence gives the degree map:
:$\deg: \operatorname(X) \to \operatorname^2(X; \mathbb) \simeq \mathbb^r.$
By definition, the Jacobian variety ''J''(''X'') of ''X'' is the identity component of the kernel of this map. Then the previous exact sequence gives:
:$0 \to \mathbb^* \to (\mathbb^*)^r \to \Gamma(\widetilde, \mathcal) \to J(X) \to J(\widetilde) \to 0.$

We next define the dual graph (algebraic curve), dual graph of ''X''; a one-dimensional CW complex defined as follows. (related to whether a curve is of compact type or not)

The Jacobian of a curve

Let ''C'' be a smooth connected curve. Given an integer ''d'', let $\operatorname^d C$ denote the set of isomorphism classes of line bundles on ''C'' of degree ''d''. It can be shown to have a structure of an algebraic variety.

For each integer ''d'' > 0, let $C^d, C_d$ denote respectively the ''d''-th fold Cartesian and symmetric product of ''C''; by definition, $C_d$ is the quotient of $C^d$ by the symmetric group permuting the factors.

Fix a base point $p_0$ of ''C''. Then there is the map
:$u: C_d \to J(C).$

Stable bundles on a curve

The Jacobian of a curve can be generalized to higher-rank vector bundles; a key notion introduced by Mumford that allows for a moduli construction is that of stability.

Let ''C'' be a connected smooth curve. A rank-2 vector bundle ''E'' on ''C'' is said to be ''stable'' if for every line subbundle ''L'' of ''E'',
:$\operatorname L < \operatorname E$.

Given some line bundle ''L'' on ''C'', let $SU_C(2, L)$ denote the set of isomorphism classes of rank-2 stable bundles ''E'' on ''C'' whose determinants are isomorphic to ''L''.

Generalization: $\operatorname_G(C)$

The osculating behavior of a curve

Vanishing sequence

Given a linear series ''V'' on a curve ''X'', the image of it under $\operatorname_p$ is a finite set and following the tradition we write it as
:$a_0(V, p) < a_1(V, p) < \cdots < a_r(V, p).$
This sequence is called the vanishing sequence. For example, $a_0(V, p)$ is the multiplicity of a base point ''p''. We think of higher $a_i(V, p)$ as encoding information about inflection of the Kodaira map $\varphi_V$. The ramification sequence is then
:$b_i(V, p) = a_i(V, p) - i.$
Their sum is called the ramification index of ''p''. The global ramification is given by the following formula:

Bundle of principal parts

Uniformization

An elliptic curve ''X'' over the complex numbers has a uniformization
$\mathbb \to X$
given by taking the quotient by a lattice.

Relative curve

A relative curve or a curve over a scheme ''S'' or a relative curve is a flat morphism, flat morphism of schemes $X \to S$ such that each geometric fiber is an algebraic curve; in other words, it is a family of curves parametrized by the base scheme ''S''.

See also Semistable reduction theorem.

The Mumford–Tate uniformization

This generalizes the classical construction due to Tate (cf. Tate curve) Given a smooth projective curve of genus at least two and has a split degeneration.

See also

*Severi variety (Hilbert scheme)
*Hurwitz scheme

Notes

References

*E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris, ''Geometry of algebraic curves.'' Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 267, Springer-Verlag, New York, 1985. MR0770932
*E. Arbarello, M. Cornalba, and P.A. Griffiths, ''Geometry of algebraic curves.'' Vol. II, with a contribution by Joseph Daniel Harris, Grundlehren der Mathematischen Wissenschaften, vol. 268, Springer, Heidelberg, 2011. MR-2807457
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*Mumford, D.: An analytic construction of degenerating curves over complete local rings. Compos. Math. 24, 129–174 (1972)
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Further reading

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*{{cite web, title=Riemann Surfaces §4.2.3 The Riemann surface of an algebraic function , url=http://www.math.tifr.res.in/~publ/pamphlets/riemann.pdf

Algebraic curves