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In algebraic geometry, a '''complete algebraic curve''' is an [[algebraic curve]] that is [[complete variety|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.<ref>{{harvnb|Hartshorne|loc=Ch. III., Exercise 5.8.}}</ref> 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 <math>\mathbb{P}^3</math> is called an (algebraic) [[space curve]], while a curve in <math>\mathbb{P}^2</math> is called a plane curve. By means of a [[projection from a point]], any smooth complete or projective curve can be embedded into <math>\mathbb{P}^3</math>;<ref>{{harvnb|Hartshorne|loc=Ch. IV., Corollay 3.6.}}</ref> thus, up to a projection, every (smooth) curve is a space curve. Up to a birational morphism, every curve can be embedded into <math>\mathbb{P}^2</math> as a [[nodal curve]].<ref>{{harvnb|Hartshorne|loc=Ch. IV., Theorem 3.10.}}</ref>

[[Riemann's existence theorem]] says that the category of [[compact Riemann surface]]s 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.<ref>{{harvnb|Hartshorne|1977|loc=Ch. I, § 6.}}</ref> Let <math>C_K</math> denote the set of all [[discrete valuation ring]]s of <math>K/k</math>. We put the topology on <math>C_K</math> so that the closed subsets are either finite subsets or the whole space. We then make it a [[locally ringed space]] by taking <math>\mathcal{O}(U)</math> to be the intersection <math>\cap_{R \in U} R</math>. Then the <math>C_K</math> for various function fields ''K'' of transcendence degree one form a category that is equivalent to the category of smooth projective curves.<ref>{{harvnb|Hartshorne|1977|loc=Ch. I, § 6. Corollary 6.12.}}</ref>

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

== Smooth completion of an affine curve ==
{{expand section}}
Let <math>C_0 = V(f) \subset \mathbb{A}^2</math> be a smooth affine curve given by a polynomial ''f'' in two variables. The closure <math>\overline{C_0}</math> in <math>\mathbb{P}^2</math>, the [[projective completion]] of it, may or may not be smooth. The normalization ''C'' of <math>\overline{C_0}</math> is smooth and contains <math>C_0</math> as an open dense subset. Then the curve <math>C</math> is called the [[smooth completion]] of <math>C_0</math>.<ref name="smooth completion">{{harvnb|ACGH|1985|loc=Ch I, Exercise A.}}</ref> (Note the smooth completion of <math>C_0</math> is unique up to isomorphism since two smooth curves are isomorphic if they are birational to each other.)

For example, if <math>f = y^2 - x^3 + 1</math>, then <math>\overline{C_0}</math> is given by <math>y^2 z = x^3 - z^3</math>, which is smooth (by a Jacobian computation). On the other hand, consider <math>f = y^2 - x^6 + 1</math>. Then, by a Jacobian computation, <math>\overline{C_0}</math> is not smooth. In fact, <math>C_0</math> 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 <math>y(x)</math> when <math>f(x, y(x)) \equiv 0</math>.<ref name="smooth completion" />. Conversely, each compact Riemann surface is of that form;{{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:
:<math>f: C - B \to \mathbb{P}(V^*)</math>
that maps each point <math>P</math> in <math>C - B</math> to the hyperplane <math>\{ s \in V | s(P) = 0 \}</math>. 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 <math>|K| = \mathbb{P}(\Gamma(C, \omega_C))</math> and the corresponding map is called the canonical map.

Let <math>g</math> be the genus of a smooth curve ''C''. If <math>g = 0</math>, then <math>|K|</math> is empty while if <math>g = 1</math>, then <math>|K| = 0</math>. If <math>g \ge 2</math>, then the canonical linear system <math>|K|</math> can be shown to have no base point and thus determines the morphism <math>f : C \to \mathbb{P}^{g-1}</math>. 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 <math>\mathbb{P}^3</math> ==
The classification of a smooth projective curve begins with specifying a genus. For genus zero, there is only one: the projective line <math>\mathbb{P}^1</math> (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:<ref>{{harvnb|Hartshorne|loc=Ch. IV., Exercise 2.2.}}</ref>
*Each genus-two curve ''X'' comes with the map <math>f: X \to \mathbb{P}^1</math> 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 <math>\ge 3</math>, the following terminology is used;
*Given a smooth curve ''C'', a divisor ''D'' on it and a vector subspace <math>V \subset H^0(C, \mathcal{O}(D))</math>, one says the linear system <math>\mathbb{P}(V)</math> is a g<sup>r</sup><sub>d</sub> if ''V'' has dimension ''r''+1 and ''D'' has degree ''d''. One says ''C'' has a g<sup>r</sup><sub>d</sub> if there is such a linear system.

== Specific curves ==

=== Canonical curve ===
{{empty section|date=July 2022}}

=== Stable curve ===
A [[stable curve]] is a connected nodal curve with finite automorphism group.<!--
{{math theorem|name=Proposition|Let ''X'' be a connected curve with δ nodes and ν irreducible components, each having genus ''g''<sub>''i''</sub>. Then ''X'' admits a dualizing sheaf and the arithmetic genus ''g'' of ''X'' is given by
:<math>g = \sum_1^\nu g_i - \nu + 1 - \delta.</math>}}
Proof: Let <math>\widetilde{X}_i</math> be the normalization of an irreducible component <math>X_i</math> of ''X''. Then, by the [[Riemann–Roch theorem|Riemann–Roch formula]],
:<math>\chi(\mathcal{O}_X) = \sum_1^\nu \chi(\mathcal{O}_{\widetilde{X}_i}) + \delta = \sum_{i=1}^\nu (1 - g_i) + \delta.</math>--><!-- We need to expand the below 
== Automorphism group ==
The automorphism group of a smooth curve of genus ''g'' has order at most 84(''g''&nbsp;−&nbsp;1).-->

=== Spectral curve ===
{{empty section|date=July 2022}}

== Vector bundles on a curve ==
=== Line bundles and dual graph ===
Let ''X'' be a possibly singular curve. Then
:<math>0 \to \mathbb{C}^* \to (\mathbb{C}^*)^r \to \Gamma(X, \mathcal{F}) \to \operatorname{Pic}(X) \to \operatorname{Pic}(\widetilde{X}) \to 0.</math>
where ''r'' is the number of irreducible components of ''X'', <math>\pi:\widetilde{X} \to X</math> is the [[normal scheme|normalization]] and <math>\mathcal{F} = \pi_* \mathcal{O}_{\widetilde{X}}/\mathcal{O}_X</math>. (To get this use the fact <math>\operatorname{Pic}(X) = \operatorname{H}^1(X, \mathcal{O}_X^*)</math> and <math>\operatorname{Pic}(\widetilde{X}) = \operatorname{H}^1(\widetilde{X}, \mathcal{O}_{\widetilde{X}}^*) = \operatorname{H}^1(X, \pi_* \mathcal{O}_{\widetilde{X}}^*).</math>)

Taking the long exact sequence of the [[exponential sheaf sequence]] gives the degree map:
:<math>\deg: \operatorname{Pic}(X) \to \operatorname{H}^2(X; \mathbb{Z}) \simeq \mathbb{Z}^r.</math>
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:
:<math>0 \to \mathbb{C}^* \to (\mathbb{C}^*)^r \to \Gamma(\widetilde{X}, \mathcal{F}) \to J(X) \to J(\widetilde{X}) \to 0.</math>

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 <math>\operatorname{Pic}^d C</math> 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 <math>C^d, C_d</math> denote respectively the ''d''-th fold Cartesian and [[symmetric product]] of ''C''; by definition, <math>C_d</math> is the quotient of <math>C^d</math> by the symmetric group permuting the factors.

Fix a base point <math>p_0</math> of ''C''. Then there is the map 
:<math>u: C_d \to J(C)</math>
given by <math>(q_1, \dots, q_d) \mapsto</math>.

=== Stable bundles on a curve ===
{{main|stable bundle}}

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'',
:<math>\operatorname{deg} L < {1 \over 2} \operatorname{deg} E</math>.

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

=== Generalization: <math>\operatorname{Bun}_G(C)</math> ===
{{main|moduli stack of principal bundles}}

== The osculating behavior of a curve ==

=== Vanishing sequence ===
Given a linear series ''V'' on a curve ''X'', the image of it under <math>\operatorname{ord}_p</math> is a finite set and following the tradition we write it as
:<math>a_0(V, p) < a_1(V, p) < \cdots < a_r(V, p).</math>
This sequence is called the vanishing sequence. For example, <math>a_0(V, p)</math> is the multiplicity of a base point&nbsp;''p''. We think of higher <math>a_i(V, p)</math> as encoding information about inflection of the [[Kodaira map]] <math>\varphi_V</math>. The ramification sequence is then
:<math>b_i(V, p) = a_i(V, p) - i.</math>
Their sum is called the ramification index of ''p''. The global ramification is given by the following formula:
{{math theorem|name=[[Plücker formula]]|
:<math>\sum_{p \in X} \sum_0^r b_i(V, p) = (r+1)(d + r(g-1)).</math>}}

=== Bundle of principal parts ===
{{main|Bundle of principal parts}}

== Uniformization ==
{{expand section}}
An elliptic curve ''X'' over the complex numbers has a uniformization
<math>\mathbb{C} \to X</math>
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]] <math>X \to S</math> 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]])<ref>{{cite book | url=https://books.google.com/books?id=bTJ8CwAAQBAJ&pg=PR3 | title=Schottky Groups and Mumford Curves | isbn=9783540383048 | last1=Gerritzen | first1=L. | last2=Van Der Put | first2=M. | date=14 November 2006 | publisher=Springer }}</ref> In {{harvnb|Mumford|1972}}, 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]]

== Notes ==
{{reflist}}

== 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 AG}}
*{{cite book
| last        = Mukai
| first       = S.
| year        = 2002
| title       = An introduction to invariants and moduli
| series      = Cambridge Studies in Advanced Mathematics
| volume      = 81
| isbn        = 978-0-521-80906-1
| url         = http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521809061
}}
*Mumford, D.: An analytic construction of degenerating curves over complete local rings. Compos. Math. 24, 129–174 (1972)
*{{cite journal |last1=McMcallum |first1=W. |last2=Poonen |first2=B. |title= The method of Chabauty and Coleman |volume=32 |date=2012 |journal=Panoramas et Synthèses |pages= 99–117}}
*{{cite book |last1=Shimura |first1=Gorō |title=Introduction to the Arithmetic Theory of Automorphic Functions |date=21 August 1971 |publisher=Princeton University Press |isbn=978-0-691-08092-5 |url= https://books.google.com/books?id=-PFtGa9fZooC}}
*{{cite arXiv |last1=Voight |first1=John |last2=Zureick-Brown |first2=David |title=The canonical ring of a stacky curve |date=16 March 2022 |class=math.AG |eprint=1501.04657 }}

== Further reading ==
* {{cite book
 | last1=Griffiths | first1=Phillip | authorlink1=Phillip Griffiths
 | last2=Harris | first2=Joe | authorlink2=Joe Harris (mathematician)
 | year = 1994
 | title = Principles of Algebraic Geometry
 | publisher = [[Wiley-Interscience]]
 | isbn = 978-0-471-05059-9
 | zbl = 0836.14001
}}
*{{cite book |url={{Google books|CiXyBwAAQBAJ|page=16|plainurl=yes}} |doi=10.1007/978-3-0348-8617-8_4 |chapter=The Riemann Surface of an Algebraic Function |title=Compact Riemann Surfaces |year=1992 |last1=Narasimhan |first1=Raghavan |pages=15–16 |isbn=978-3-7643-2742-2 }}
*{{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}}
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