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 is called an (algebraic) space curve, while a curve in is called a plane curve. By means of a projection from a point, any smooth complete or projective curve can be embedded into ;[2] thus, up to a projection, every (smooth) curve is a space curve. Up to a birational morphism, every curve can be embedded into 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
editLet 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 denote the set of all discrete valuation rings of . We put the topology on so that the closed subsets are either finite subsets or the whole space. We then make it a locally ringed space by taking to be the intersection . Then the 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 , which corresponds to a projective smooth curve.)
Smooth completion of an affine curve
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Let be a smooth affine curve given by a polynomial f in two variables. The closure in , the projective completion of it, may or may not be smooth. The normalization C of is smooth and contains as an open dense subset. Then the curve is called the smooth completion of .[6] (Note the smooth completion of is unique up to isomorphism since two smooth curves are isomorphic if they are birational to each other.)
For example, if , then is given by , which is smooth (by a Jacobian computation). On the other hand, consider . Then, by a Jacobian computation, is not smooth. In fact, 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 when .[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
editTo 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:
that maps each point in to the hyperplane . 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 and the corresponding map is called the canonical map.
Let be the genus of a smooth curve C. If , then is empty while if , then . If , then the canonical linear system can be shown to have no base point and thus determines the morphism . 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
editThe classification of a smooth projective curve begins with specifying a genus. For genus zero, there is only one: the projective line (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 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 , the following terminology is used;
- Given a smooth curve C, a divisor D on it and a vector subspace , one says the linear system is a grd if V has dimension r+1 and D has degree d. One says C has a grd if there is such a linear system.
Specific curves
editCanonical curve
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Stable curve
editA stable curve is a connected nodal curve with finite automorphism group.
Spectral curve
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Vector bundles on a curve
editLine bundles and dual graph
editLet X be a possibly singular curve. Then
where r is the number of irreducible components of X, is the normalization and . (To get this use the fact and )
Taking the long exact sequence of the exponential sheaf sequence gives the degree map:
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:
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
editLet C be a smooth connected curve. Given an integer d, let 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 denote respectively the d-th fold Cartesian and symmetric product of C; by definition, is the quotient of by the symmetric group permuting the factors.
Fix a base point of C. Then there is the map
given by .
Stable bundles on a curve
editThe 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,
- .
Given some line bundle L on C, let denote the set of isomorphism classes of rank-2 stable bundles E on C whose determinants are isomorphic to L.
Generalization:
editThe osculating behavior of a curve
editVanishing sequence
editGiven a linear series V on a curve X, the image of it under is a finite set and following the tradition we write it as
This sequence is called the vanishing sequence. For example, is the multiplicity of a base point p. We think of higher as encoding information about inflection of the Kodaira map . The ramification sequence is then
Their sum is called the ramification index of p. The global ramification is given by the following formula:
Bundle of principal parts
editUniformization
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An elliptic curve Xover the complex numbers has a uniformization given by taking the quotient by a lattice.
Relative curve
editA relative curve or a curve over a scheme S or a relative curve is a flat morphism of schemes 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
editThis generalizes the classical construction due to Tate (cf. Tate curve)[8] In Mumford 1972 , Mumford showed: given a smooth projective curve of genus at least two and has a split degeneration,
See also
editNotes
edit- ^ 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.
- ^ Gerritzen, L.; Van Der Put, M. (14 November 2006). Schottky Groups and Mumford Curves. Springer. ISBN 9783540383048.
References
edit- 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. (2017). "THE METHOD OF CHABAUTY AND COLEMAN". S2CID 41964616.
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(help) - Voight, John; Zureick-Brown, David (16 March 2022). "The canonical ring of a stacky curve". arXiv:1501.04657 [math.AG].
Further reading
edit- 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).