Draft: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 ${\mathbb{P}}^3$ is called an (algebraic) space curve, while a curve in ${\mathbb{P}}^2$ is called a plane curve. By means of a projection from a point, any smooth complete or projective curve can be embedded into ${\mathbb{P}}^3$; thus, up to a projection, every (smooth) curve is a space curve. Up to a birational morphism, every curve can be embedded into ${\mathbb{P}}^2$ 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 $𝒪(U)$ to be the intersection ${\cap }_{R\in U}R$. 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

Template:Expand section Let $C_0=V(f)\subset {𝔸}^2$ be a smooth affine curve given by a polynomial f in two variables. The closure $\overline{C_0}$ in ${\mathbb{P}}^2$, the projective completion of it, may or may not be smooth. The normalization C of $\overline{C_0}$ 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 $\overline{C_0}$ 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, $\overline{C_0}$ 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;Template:Fact 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:

$f:C-B\to \mathbb{P}(V^{*})$

that maps each point $P$ in $C-B$ to the hyperplane $\{s\in V|s(P)=0\}$. 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 $|K|=\mathbb{P}(\mathrm{\Gamma }(C,{\omega }_C))$ 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 $f:C\to {\mathbb{P}}^{g-1}$. 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 ${\mathbb{P}}^3$

The classification of a smooth projective curve begins with specifying a genus. For genus zero, there is only one: the projective line ${\mathbb{P}}^1$ (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 $f:X\to {\mathbb{P}}^1$ determined by the canonical divisor; called the canonical map. The canonical map has exactly 6 ramified points of index 2.
• Conversely, given 6 points,

For genus $\ge 3$, the following terminology is used;

• Given a smooth curve C, a divisor D on it and a vector subspace $V\subset H^0(C,𝒪(D))$, one says the linear system $\mathbb{P}(V)$ 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.

Specific curves

Stable curve

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

Vector bundles on a curve

Line bundles and dual graph

Let X be a possibly singular curve. Then

$0\to {ℂ}^{*}\to ({ℂ}^{*}{)}^r\to \mathrm{\Gamma }(X,ℱ)\to \mathrm{Pic}(X)\to \mathrm{Pic}(\tilde{X})\to 0.$

where r is the number of irreducible components of X, $\pi :\tilde{X}\to X$ is the normalization and $ℱ={\pi }_{*}{𝒪}_{\tilde{X}}/{𝒪}_X$. (To get this use the fact $\mathrm{Pic}(X)={\mathrm{H}}^1(X,{𝒪}_X^{*})$ and $\mathrm{Pic}(\tilde{X})={\mathrm{H}}^1(\tilde{X},{𝒪}_{\tilde{X}}^{*})={\mathrm{H}}^1(X,{\pi }_{*}{𝒪}_{\tilde{X}}^{*}).$)

Taking the long exact sequence of the exponential sheaf sequence gives the degree map:

$\mathrm{deg}:\mathrm{Pic}(X)\to {\mathrm{H}}^2(X;\mathbb{Z})\simeq {\mathbb{Z}}^r.$

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:

$0\to {ℂ}^{*}\to ({ℂ}^{*}{)}^r\to \mathrm{\Gamma }(\tilde{X},ℱ)\to J(X)\to J(\tilde{X})\to 0.$

We next define the 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 ${\mathrm{Pic}}^{d}C$ 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

$u:C_d\to J(C)$

given by $(q_1,\dots ,q_d)\mapsto$.

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,

$\mathrm{deg}L<\frac{1}{2}\mathrm{deg}E$.

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

The osculating behavior of a curve

Vanishing sequence

Given a linear series V on a curve X, the image of it under ${\mathrm{ord}}_p$ 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 ${\phi }_V$. 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: Template:Math theorem

Uniformization

Template:Expand section An elliptic curve X over the complex numbers has a uniformization $ℂ\to X$ 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 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) In Template:Harvnb, Mumford showed: given a smooth projective curve of genus at least two and has a split degeneration,

See also

• Severi variety (Hilbert scheme)
• Hurwitz scheme

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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