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. ^[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 ${\displaystyle \mathbb {P} ^{3}}$ is called an (algebraic) space curve, while a curve in ${\displaystyle \mathbb {P} ^{2}}$ is called a plane curve. By means of a projection from a point, any smooth projective curve can be embedded into ${\displaystyle \mathbb {P} ^{3}}$; ^[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 ${\displaystyle \mathbb {P} ^{2}}$ 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 ^{[ disambiguation needed ]}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 ${\displaystyle C_{K}}$ denote the set of all discrete valuation rings of ${\displaystyle K/k}$. We put the topology on ${\displaystyle C_{K}}$ so that the closed subsets are either finite subsets or the whole space. We then make it a locally ringed space by taking ${\displaystyle {\mathcal {O}}(U)}$ to be the intersection ${\displaystyle \cap _{R\in U}R}$. Then the ${\displaystyle C_{K}}$ 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 ${\displaystyle C_{K},K=k(C)}$, which corresponds to a projective smooth curve.)

Smooth completion of an affine curve

Let ${\displaystyle C_{0}=V(f)\subset \mathbb {A} ^{2}}$ be a smooth affine curve given by a polynomial f in two variables. The closure ${\displaystyle {\overline {C_{0}}}}$ in ${\displaystyle \mathbb {P} ^{2}}$, the projective completion of it, may or may not be smooth. The normalization C of ${\displaystyle {\overline {C_{0}}}}$ is smooth and contains ${\displaystyle C_{0}}$ as an open dense subset. Then the curve ${\displaystyle C}$ is called the smooth completion of ${\displaystyle C_{0}}$. ^[6] (Note the smooth completion of ${\displaystyle C_{0}}$ is unique up to isomorphism since two smooth curves are isomorphic if they are birational to each other.)

For example, if ${\displaystyle f=y^{2}-x^{3}+1}$, then ${\displaystyle {\overline {C_{0}}}}$ is given by ${\displaystyle y^{2}z=x^{3}-z^{3}}$, which is smooth (by a Jacobian computation). On the other hand, consider ${\displaystyle f=y^{2}-x^{6}+1}$. Then, by a Jacobian computation, ${\displaystyle {\overline {C_{0}}}}$ is not smooth. In fact, ${\displaystyle C_{0}}$ 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 ${\displaystyle y(x)}$ when ${\displaystyle f(x,y(x))\equiv 0}$. ^[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

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:

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

that maps each point ${\displaystyle P}$ in ${\displaystyle C-B}$ to the hyperplane ${\displaystyle \{s\in V|s(P)=0\}}$. 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 ${\displaystyle |K|=\mathbb {P} (\Gamma (C,\omega _{C}))}$ and the corresponding map is called the canonical map.

Let ${\displaystyle g}$ be the genus of a smooth curve C. If ${\displaystyle g=0}$, then ${\displaystyle |K|}$ is empty while if ${\displaystyle g=1}$, then ${\displaystyle |K|=0}$. If ${\displaystyle g\geq 2}$, then the canonical linear system ${\displaystyle |K|}$ can be shown to have no base point and thus determines the morphism ${\displaystyle f:C\to \mathbb {P} ^{g-1}}$. 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 ^{[ disambiguation needed ]} 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 ${\displaystyle \mathbb {P} ^{3}}$

The classification of a smooth projective curve begins with specifying a genus. For genus zero, there is only one: the projective line ${\displaystyle \mathbb {P} ^{1}}$ (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 ${\displaystyle f:X\to \mathbb {P} ^{1}}$ 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 ${\displaystyle a_{1},\dots ,a_{6}}$, let ${\displaystyle K}$ be the field extension of ${\displaystyle k(x)}$, x a variable, given by the equation ${\displaystyle y^{2}=(x-a_{1})\cdots (x-a_{6})}$ and ${\displaystyle f:X\to \mathbb {P} ^{1}}$ the map corresponding to the extension. Then ${\displaystyle X}$ is a genus-two curve and ${\displaystyle f}$ ramifies exactly over those six points.

For genus ${\displaystyle \geq 3}$, the following terminology is used:^{[ citation needed ]}

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

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

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

Specific curves

Canonical curve

If X is a nonhyperelliptic curve of genus ${\displaystyle \geq 3}$, then the linear system ${\displaystyle |K|}$ 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

For genus ${\displaystyle g\geq 2}$, a stable curve is a connected nodal curve with finite automorphism group.^{[ citation needed ]}

Vector bundles on a curve

Line bundles and dual graph

Let X be a possibly singular curve over complex numbers. Then

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

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

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

${\displaystyle \deg$ :\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:

${\displaystyle 0\to \mathbb {C} ^{*}\to (\mathbb {C} ^{*})^{r}\to \Gamma ({\widetilde {X}},{\mathcal {F}})\to J(X)\to J({\widetilde {X}})\to 0.}$

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 ${\displaystyle \operatorname {Pic} ^{d}C}$ 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 ${\displaystyle C^{d},C_{d}}$ denote respectively the d-th fold Cartesian and symmetric product ^{[ disambiguation needed ]} of C; by definition, ${\displaystyle C_{d}}$ is the quotient of ${\displaystyle C^{d}}$ by the symmetric group permuting the factors.

Fix a base point ${\displaystyle p_{0}}$ of C. Then there is the map

${\displaystyle u:C_{d}\to J(C).}$

Stable bundles on a curve

Main article: 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,

${\displaystyle \operatorname {deg} L<{1 \over 2}\operatorname {deg} E}$.

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

Generalization: ${\displaystyle \operatorname {Bun} _{G}(C)}$

Main article: 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 ${\displaystyle \operatorname {ord} _{p}}$ is a finite set and following the tradition we write it as

${\displaystyle a_{0}(V,p)<a_{1}(V,p)<\cdots <a_{r}(V,p).}$

This sequence is called the vanishing sequence. For example, ${\displaystyle a_{0}(V,p)}$ is the multiplicity of a base point p. We think of higher ${\displaystyle a_{i}(V,p)}$ as encoding information about inflection of the Kodaira map ${\displaystyle \varphi _{V}}$. The ramification sequence is then

${\displaystyle b_{i}(V,p)=a_{i}(V,p)-i.}$

Their sum is called the ramification index of p. The global ramification is given by the following formula:

Plücker formula —

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

Bundle of principal parts

Main article: Bundle of principal parts

Uniformization

An elliptic curve X over the complex numbers has a uniformization ${\displaystyle \mathbb {C} \to X}$ given by taking the quotient by a lattice.^{[ citation needed ]}

Relative curve

A relative curve or a curve over a schemeS or a relative curve is a flat morphism of schemes ${\displaystyle X\to S}$ such that each geometric fiber is an algebraic curve; in other words, it is a family of curves parametrized by the base scheme S.^{[ citation needed ]}

See also Semistable reduction theorem.

The Mumford–Tate uniformization

This generalizes the classical construction due to Tate (cf. Tate curve) ^[10] Given a smooth projective curve of genus at least two and has a split degeneration. ^[11]

See also

• Severi variety (Hilbert scheme)
• Hurwitz scheme

Notes

1. Hartshorne 1977 , Ch. III., Exercise 5.8.
2. Hartshorne 1977 , Ch. IV., Corollay 3.6.
3. Hartshorne 1977 , Ch. IV., Theorem 3.10.
4. Hartshorne 1977 , Ch. I, § 6.
5. Hartshorne 1977 , Ch. I, § 6. Corollary 6.12.
6. Arbarello et al. 1985 , Ch I, Exercise A.
7. Hartshorne 1977 , Ch. IV., Exercise 2.2.
8. Hartshorne 1977 , Ch. IV., Exercise 4.8.
9. Hartshorne 1977 , Ch. IV., § 5.
10. Gerritzen, L.; Van Der Put, M. (14 November 2006). Schottky Groups and Mumford Curves. Springer. ISBN   9783540383048.
11. Mumford 1972

References

• Arbarello, E.; Cornalba, M.; Griffiths, P.A.; Harris, J. (1985). Geometry of algebraic curves. Grundlehren der Mathematischen Wissenschaften, vol. 267. Vol. 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, 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, David (1972). "An analytic construction of degenerating curves over complete local rings" (PDF). Compositio Mathematica . 24 (2): 129–174.
• 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].
• Algebraic Geometry: A First Course. Springer. 11 November 2013. ISBN   978-1-4757-2189-8.

Further reading

• Griffiths, Phillip; Harris, Joe (1994). Principles of Algebraic Geometry. Wiley-Interscience. ISBN   978-0-471-05059-9. Zbl   0836.14001.
• 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.
• "Riemann Surfaces §4.2.3 The Riemann surface of an algebraic function" (PDF).