Complete algebraic curve

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

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

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 ${\mathbb{\mathbb{P}}}^2$, 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; this is known as the Riemann existence theorem.

A map from a curve to a projective space

Main article: Linear system of divisors

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:

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

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 $|K|=\mathbb{\mathbb{P}}(\mathrm{\Gamma }(C,{\omega }_C))$ 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 $f:C\to {\mathbb{\mathbb{P}}}^{g-1}$. 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 ${\mathbb{\mathbb{P}}}^3$

The classification of a smooth projective curve begins with specifying a genus. For genus zero, there is only one: the projective line ${\mathbb{\mathbb{P}}}^1$ (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 $f:X\to {\mathbb{\mathbb{P}}}^1$ 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 $f:X\to {\mathbb{\mathbb{P}}}^1$ the map corresponding to the extension. Then $X$ is a genus-two curve and $f$ ramifies exactly over those six points.

For genus $\ge 3$, 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 $\mathbb{\mathbb{P}}(V)$ 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. Then the étale fundamental group of X is defined as:

${\pi }_1(X)=\underset{L/K}{\underleftarrow{\lim }}\mathrm{Gal}(L/K)$

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

For genus $g\ge 2$, 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 over complex numbers. Then

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

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:

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

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:

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

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

$u:C_d\to J(C).$

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:

Plücker formula — 

$\sum _{p\in X}\sum _0^r{b}_i(V,p)=(r+1)(d+r(g-1)).$

Bundle of principal parts

Main article: 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.

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

References

• Arbarello, E.; Cornalba, M.; Griffiths, P.A.; Harris, J. (1985). Geometry of algebraic curves. Grundlehren der Mathematischen Wissenschaften, vol. 267. I. New York: Springer-Verlag. 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, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9 
• Mukai, S. (2002). An introduction to invariants and moduli. Cambridge Studies in Advanced Mathematics. 81. ISBN 978-0-521-80906-1. http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521809061. 
• Mumford, David (1972). "An analytic construction of degenerating curves over complete local rings". Compositio Mathematica 24 (2): 129–174. https://www.numdam.org/article/CM_1972__24_2_129_0.pdf. 
• 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. https://books.google.com/books?id=-PFtGa9fZooC. 
• Voight, John; Zureick-Brown, David (16 March 2022). "The canonical ring of a stacky curve". arXiv:1501.04657 [math.AG].
• Algebraic Geometry: A First Course. Springer. 11 November 2013. ISBN 978-1-4757-2189-8. https://books.google.com/books?id=U-UlBQAAQBAJ&pg=281. 

Further reading

• Griffiths, Phillip; Harris, Joe (1994). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 978-0-471-05059-9. 
• Narasimhan, Raghavan (1992). "The Riemann Surface of an Algebraic Function". Compact Riemann Surfaces. pp. 15–16. doi:10.1007/978-3-0348-8617-8_4. ISBN 978-3-7643-2742-2. https://books.google.com/books?id=CiXyBwAAQBAJ&pg=PA16. 
• "Riemann Surfaces §4.2.3 The Riemann surface of an algebraic function". http://www.math.tifr.res.in/~publ/pamphlets/riemann.pdf.