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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.^[1] 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 projective curve can be embedded into $\mathbb {P} ^{3}$ ;^[2] thus, up to a projection, every (smooth) curve is a space curve. Up to a birational morphism, every (smooth) curve can be embedded into $\mathbb {P} ^{2}$ as a nodal curve.^[3]

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.^[4] 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 ${\mathcal {O}}(U)$ 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.^[5]

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 $C_{0}=V(f)\subset \mathbb {A} ^{2}$ 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 $C_{0}$ .^[6] (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 $f(x,y(x))\equiv 0$ .^[6]. Conversely, each compact Riemann surface is of that form;^{[citation needed]} 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 $|K|=\mathbb {P} (\Gamma (C,\omega _{C}))$ 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 $g\geq 2$ , 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:^[7]

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 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 $f:X\to \mathbb {P} ^{1}$ the map corresponding to the extension. Then $X$ is a genus-two curve and $f$ ramifies exactly over those six points.

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

Given a smooth curve C, a divisor D on it and a vector subspace $V\subset H^{0}(C,{\mathcal {O}}(D))$ , 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.

Fundamental group

Let X be a smooth complete algebraic curve.^{[clarification needed]} Then the étale fundamental group of X is defined as:

\pi _{1}(X)=\varprojlim _{L/K}\operatorname {Gal} (L/K)

where $K$ is the function field of X and $L/K$ is a Galois extension.^[8]

Specific curves

Canonical curve

If X is a nonhyperelliptic curve of genus $\geq 3$ , 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.^[9]

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 {C} ^{*}\to (\mathbb {C} ^{*})^{r}\to \Gamma (X,{\mathcal {F}})\to \operatorname {Pic} (X)\to \operatorname {Pic} ({\widetilde {X}})\to 0.

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

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

\deg :\operatorname {Pic} (X)\to \operatorname {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 \mathbb {C} ^{*}\to (\mathbb {C} ^{*})^{r}\to \Gamma ({\widetilde {X}},{\mathcal {F}})\to J(X)\to J({\widetilde {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 $\operatorname {Pic} ^{d}C$ 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).

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 {deg} L<{1 \over 2}\operatorname {deg} 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: $\operatorname {Bun} _{G}(C)$

The osculating behavior of a curve

Vanishing sequence

Given a linear series V on a curve X, the image of it under $\operatorname {ord} _{p}$ 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 $\varphi _{V}$ . 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:

Plücker formula—

\sum _{p\in X}\sum _{0}^{r}b_{i}(V,p)=(r+1)(d+r(g-1)).

Bundle of principal parts

Uniformization

An elliptic curve X over the complex numbers has a uniformization $\mathbb {C} \to X$ 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.

The Mumford–Tate uniformization

This generalizes the classical construction due to Tate (cf. Tate curve)^[10] In Mumford 1972, Mumford showed: given a smooth projective curve of genus at least two and has a split degeneration,

Notes

^ Hartshorne, Ch. III., Exercise 5.8.
^ Hartshorne, Ch. IV., Corollay 3.6.
^ Hartshorne, Ch. IV., Theorem 3.10.
^ Hartshorne 1977, Ch. I, § 6.
^ Hartshorne 1977, Ch. I, § 6. Corollary 6.12.
^ ^a ^b ACGH 1985, Ch I, Exercise A.
^ Hartshorne, Ch. IV., Exercise 2.2.
^ Hartshorne 1977, Ch. IV., Exercise 4.8.
^ Hartshorne 1977, Ch. IV., § 5.
^ Gerritzen, L.; Van Der Put, M. (14 November 2006). Schottky Groups and Mumford Curves. Springer. ISBN 9783540383048.

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
Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Mukai, S. (2002). An introduction to invariants and moduli. Cambridge Studies in Advanced Mathematics. Vol. 81. ISBN 978-0-521-80906-1.
Mumford, D.: An analytic construction of degenerating curves over complete local rings. Compos. Math. 24, 129–174 (1972)
McMcallum, W.; Poonen, B. (2012). "The method of Chabauty and Coleman". Panoramas et Synthèses. 32: 99–117.
Shimura, Gorō (21 August 1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press. ISBN 978-0-691-08092-5.
Voight, John; Zureick-Brown, David (16 March 2022). "The canonical ring of a stacky curve". arXiv:1501.04657 [math.AG].