Algebraic curve

An algebraic variety of dimension one. An algebraic curve is the most frequently studied object in algebraic geometry. In the sequel, an algebraic curve means an irreducible algebraic curve over an algebraically closed field.

The simplest and clearest concept is that of a plane affine algebraic curve. This is a set of points in an affine plane $ A _{k} ^{2} $ satisfying the equation $ f (x,\ y) = 0 $ , where $ f (x,\ y) $ is a polynomial with coefficients from an algebraically closed field $ k $ . The field of rational functions of an irreducible algebraic curve over $ k $ is the field of algebraic functions in one variable of the form $ k (x,\ y) $ , where $ x $ and $ y $ are connected by the equation $ f (x,\ y) = 0 $ , where $ f (x,\ y) $ is a polynomial over $ k $ . This means that every algebraic curve is birationally isomorphic to a plane affine curve.

It has been known for a long time that even when studying affine curves fundamental relationships can only be revealed by considering points at an infinite distance, and by a detailed study of the singular points. In order to study all the points of an affine curve, the curve is imbedded into a projective space $ P ^{n} $ , with subsequent closure in the Zariski topology. In this way a projective curve $ X $ is obtained, and the initial affine curve $ Y $ may be obtained from $ X $ by deleting a finite number of points. If $ Y $ is irreducible, then $ X $ and $ Y $ are birationally isomorphic. Every complete algebraic curve is projective. If $ X $ is a smooth projective curve, then all valuation rings of the field $ k(x) $ are given by the local rings $ {\mathcal O} _{x} $ , $ x \in X $ ( cf. Local ring). If two smooth projective curves are birationally equivalent, then they are isomorphic. A normal algebraic curve is smooth. In particular, any irreducible algebraic curve is birationally equivalent to a smooth projective curve. The projective model of an algebraic curve, obtained by the process of normalization, lies in some space $ P ^{n} $ . Any smooth projective curve is isomorphic to a curve situated in $ P ^{3} $ . Any plane algebraic curve can be converted by a Cremona transformation into a curve with ordinary singular points.

Divisors on a smooth algebraic curve are represented by linear combinations of points with integer coefficients $$ D = \sum _ {x \in X} n _{x} x , n _{x} \in \mathbf Z , $$ where $ n _{x} = 0 $ for almost all $ x $ ( cf. Divisor). If all $ n _{x} \geq 0 $ , the divisor $ D $ is called positive, or effective, which is written as $ D \geq 0 $ . The degree of the divisor $ D $ is the number $$ \mathop{\rm deg}\nolimits \ D = \sum _{x} n _{x} . $$ The principal divisors form a subgroup $ P(X) $ of the group $ \mathop{\rm Div}\nolimits \ X $ of all divisors on $ X $ . The quotient group $ \mathop{\rm Div}\nolimits (X)/P(X) $ is called the group of divisor classes and is denoted by $ \mathop{\rm Cl}\nolimits (X) $ . The group $ \mathop{\rm Cl}\nolimits (X) $ is isomorphic to the group $ \mathop{\rm Pic}\nolimits (X) $ of classes of one-dimensional vector bundles on $ X $ ( cf. Vector bundle, algebraic). The degree of the principal divisors on a smooth projective curve is zero, and thus all divisors in one class are of the same degree. In particular, one can speak of the degree of a divisor class, and of the subgroup $ \mathop{\rm Cl}\nolimits ^{0} (X) $ of divisor classes of degree zero. The following equality is valid: $$ \mathop{\rm Cl}\nolimits (X) / \mathop{\rm Cl}\nolimits ^{0} (X) = \mathbf Z . $$ For the line $ P ^{1} , \mathop{\rm Cl}\nolimits (P ^{1} ) = \mathbf Z , \mathop{\rm Cl}\nolimits ^{0} (P ^{1} ) = 0 $ , i.e. any divisor of degree zero is a principal divisor. This property is characteristic for rational smooth projective curves.

For any complete algebraic curve $ X $ , the number $ \pi = \mathop{\rm dim}\nolimits \ H ^{1} (X,\ {\mathcal O} _{X} ) $ is known as the arithmetic genus of the algebraic curve $ X $ . If $ X $ is smooth, $ \pi $ is identical with the dimension of the space $ H ^{0} (X,\ \Omega _{X} ^{1} ) $ of all regular differential forms on $ X $ ; this dimension is known as the genus of $ X $ . By definition, the genus of an algebraic curve is equal to the genus of its non-singular model. For any non-negative integer $ g $ there exists an algebraic curve of genus $ g $ . Rational curves are distinguished by the equality $ g = 0 $ . If $ X $ is a projective plane curve of order $ m $ , then $$ \pi = \frac{( m - 1 ) ( m - 2 )}{2} , $$ and its genus is given by the formula: $$ g = \frac{( m -1 ) ( m - 2 )}{2} - d , $$ where $ d $ is a non-negative integer which is a measure of the smoothness on $ X $ . If $ X $ has only ordinary double points, $ d $ is simply the number of singular points.

In particular, the genus of a plane smooth projective curve is given by $$ g = \frac{( m - 1 ) ( m - 2 )}{2} , $$ which means that not every smooth projective curve is plane. For a curve $ X $ in space the following estimate holds: $$ \tag{1} g \leq \left \{ \begin{array}{ll} \frac{( n - 2 ) ^{2}}{4} & \textrm{ for } \textrm{ even } n, \\ \frac{( n - 1 ) ( n - 3 )}{4} & \textrm{ for } \textrm{ odd } n, \\ \end{array} \right. $$ where $ n $ is the degree of $ X $ . Curves of degree $ n $ of maximal genus exist for each value of $ n $ and lie on a quadric (G. Halphen, 1870).

The degree of the canonical class $ K _{X} $ of a smooth projective curve $ X $ is connected with the genus of the curve by means of the formula $ \mathop{\rm deg}\nolimits \ K _{X} = 2g - 2 $ . If a smooth projective curve $ X $ lies on a smooth algebraic surface $ F $ , the adjunction formula $ K _{X} = X(X + K _{F} ) $ holds. In particular, $ \mathop{\rm deg}\nolimits \ K _{X} = (X) ^{2} + (X \cdot K _{F} ) $ . For an arbitrary divisor $ D $ on $ X $ , one can consider the subset of the field $ k(X) $ consisting of zero and of the functions $ f $ for which $ (f) + D \geq 0 $ . This is a linear space over $ k $ of finite dimension $ l(D) $ . The dimension of the complete linear system defined by the divisor $ D $ is $ l(D) - 1 $ and the calculation of $ l(D) $ is an important task of the theory of algebraic curves. The strongest relevant result is the Riemann–Roch theorem. For smooth projective curves this theorem is the equality $$ l (D) - l ( K - D ) = \mathop{\rm deg}\nolimits (D) - g + 1 , $$ where $ g $ is the genus of the curve $ X $ . If $ l(K - D) > 0 $ ( or if $ l(K - D) = 0 $ ), one says that $ D $ is special (or, respectively, non-special). For non-special divisors the Riemann–Roch theorem yields $ l(D) = \mathop{\rm deg}\nolimits (D) - g + 1 $ . Each divisor of degree higher than $ 2g - 2 $ is non-special.

The class of divisors which are linearly equivalent to a divisor $ D $ on a smooth projective curve $ X $ defines a point on the Jacobi variety $ J(X) $ of $ X $ . This variety is identical with the Albanese variety and with the Picard variety of $ X $ . Points which correspond to classes of special divisors are singular points of the Poincaré divisor on $ J(X) $ . If $ G _{n} ^{r} $ denotes the subset of points of $ J(X) $ which correspond to the classes of divisors $ D $ with $ \mathop{\rm deg}\nolimits \ D = n $ and $ l(D) = r $ , then $ G _{n} ^{r} $ forms a subscheme in $ J(X) $ and $$ \mathop{\rm dim}\nolimits \ G _{n} ^{r} \geq r ( n - r + 1 ) - ( r - 1 ) g $$ ( the Riemann–Brill–Noether theorem). This theorem has numerous applications, one of which will now be described. Any divisor $ D $ for which $ l(D) \geq 1 $ defines a rational mapping $ \phi _{D} $ of the curve $ X $ into the projective space $ P ^{l(D)-1} $ . The mapping $ \phi _{D} $ depends on the class of $ D $ . If $ \mathop{\rm deg}\nolimits (D ) \geq 2g + 1 $ , then $ \phi _{D} $ defines an isomorphic imbedding of $ X $ into $ P ^{m} $ , while $ \phi _{D} (X) $ is not contained in any proper subspace of the space $ P ^{m} $ ( $ m \equiv l(D) - 1 $ ). Mappings $ \phi $ which correspond to a multiple $ nK $ of the canonical class of $ X $ are the most interesting from the point of view of the birational classification of curves. If $ g > 1 $ , the class $ 3K $ defines an isomorphic imbedding of the smooth projective curve into $ P ^{5g-6} $ . Two curves $ X $ and $ Y $ are birationally equivalent if and only if their images $ \phi _{3K} (X) $ and $ \phi _{3K} (Y) $ are obtained from each other by a projective transformation of $ P ^{5g-6} $ . The study of the mapping $ \phi _{K} $ yielded a more precise characteristic of curves of genus $ g > 1 $ . For these curves $ \phi _{K} : \ X \rightarrow P ^{g-1} $ is an isomorphic imbedding if and only if $ X $ is not a hyper-elliptic curve. If $ \phi _{K} $ is an isomorphism, the curve $ \phi _{K} (X) $ is called canonical; it is defined uniquely up to projective transformations in $ P ^{g-1} $ . A very important task of the theory of algebraic curves is their classification up to a birational isomorphism. A number of important results have been obtained in this field, but an adequate solution of the problem is not available now (1977).

Smooth projective curves are subdivided into four classes:

1) curves of genus 0, birationally equivalent to $ P _{1} $ ;

2) curves of genus 1 (elliptic curves), birationally equivalent to a smooth cubic curve in $ P ^{2} $ ;

3) hyper-elliptic curves;

4) non-hyper-elliptic curves of genus $ g > 1 $ , birationally equivalent to a canonical curve in $ P ^{g-1} $ ( algebraic curves of basic type).

The genus of a curve does not fully characterize the birational class of an algebraic curve. The only exception are curves of genus zero. If $ k $ is the field of complex numbers $ \mathbf C $ , the set of classes of mutually-isomorphic elliptic curves is described by points in the quotient space $ H/G $ , where $ H $ is the upper half-plane and $ G $ is the modular group consisting of rational-linear transformations with integral coefficients and with determinant equal to $ + 1 $ . The space $ H/G $ has the structure of an analytic manifold isomorphic to $ \mathbf C $ ( cf. Elliptic curve). Classes of birationally equivalent curves of genus $ g > 1 $ are described by points belonging to some algebraic variety $ {\mathcal M} _{g} $ of dimension $ 3g - 3 $ , which is called the moduli variety of curves of genus $ g $ . This variety is irreducible. According to one conjecture $ {\mathcal M} _{g} $ is unirational; however, this has been proved (by F. Severi) for $ g < 11 $ only.

The following results are valid for the group $ \mathop{\rm Aut}\nolimits (X) $ of automorphisms of a smooth projective curve $ X $ . 1) If $ X $ is $ P _{k} ^{1} $ , then $ \mathop{\rm Aut}\nolimits (X) $ is the group of rational-linear transformations $ \mathop{\rm PGL}\nolimits (1,\ k ) $ . 2) If $ X $ is an elliptic curve, then $ \mathop{\rm Aut}\nolimits (X) $ is an algebraic group, the connected component of the unit of which coincides with the group of points of $ X(k) $ . 3) If $ X $ is a curve of genus $ g > 1 $ , then $ \mathop{\rm Aut}\nolimits (X) $ is always a finite group. Its order is bounded by the number $ 84(g - 1) $ [[# References|[6]]]. Weierstrass points (cf. Weierstrass point) on $ X $ play an important role in the study of the group $ \mathop{\rm Aut}\nolimits (X) $ in the latter case.

Another way of studying $ \mathop{\rm Aut}\nolimits (X) $ is based on the fact that all smooth projective curves are finite (ramified) coverings of the projective line.

Let $ X $ be a smooth projective curve defined over the field $ \mathbf C $ . The set of points of the curve $ X ( \mathbf C ) $ has the natural structure of a one-dimensional compact analytic manifold, which is also known as a compact Riemann surface. The converse is also true, i.e. any compact Riemann surface is obtained from some smooth projective curve. Usually one uses the same symbol $ X $ for the smooth projective curve and its corresponding one-dimensional complex manifold. Any connected complex manifold $ X $ can be represented as a quotient $ \widetilde{X} / G $ , where $ \widetilde{X} $ is a simply-connected complex manifold, and $ G $ is a group of automorphisms of $ \widetilde{X} $ which acts on $ \widetilde{X} $ discretely and freely. It is noteworthy that there are only three one-dimensional simply-connected connected analytic manifolds, up to an isomorphism. These are the projective line $ \mathbf C P ^{1} $ ( the Riemann sphere), the affine straight line $ \mathbf C $ ( the finite plane) and the interior of the unit disc $ D = \{ {z} : {| z | < 1} \} $ ( the Lobachevskii plane). All smooth projective curves are subdivided into three classes, depending on which one of the three types their universal covering belongs to.

The problem of classifying smooth projective curves of a given type can be reduced to the study of discrete groups of transformations of the universal coverings which act freely with a relatively compact fundamental domain. In the case of the projective line, $ G $ is the identity group; in the case of the affine straight line, $ G $ is isomorphic to a subgroup $ \Omega $ of the additive group $ \mathbf C $ which is a two-dimensional lattice in $ \mathbf C $ ; in the case of the interior of the unit disc, $ G $ is a subgroup of motions in the Lobachevskii plane which can be defined by some non-Euclidean bounded polygon. Thus, the first class above contains a unique curve $ P ^{1} $ , the second class consists of complex tori $ \mathbf C / \Omega $ that all have the structure of a one-dimensional Abelian variety (elliptic curve), the addition of points on the torus defining the group structure on the respective curve. All smooth elliptic curves are obtained in this way. The field of rational functions $ C(X) $ on an elliptic curve $ X \mathbf C / \Omega $ is isomorphic to the field of meromorphic doubly-periodic (elliptic) functions with period group $ \Omega $ . If $ f(x,\ y) = 0 $ is the equation of the affine model of the curve $ X $ , then there exists a parametrization $ x = \phi (z) $ , $ y = \psi (z) $ by elliptic functions of it (a uniformization of $ X $ ). The third class consists of all smooth projective curves $ X $ of genus $ g > 1 $ . In this case the field $ C(X) $ is isomorphic to the field of meromorphic functions on $ D $ that are invariant with respect to the group $ G $ . Such functions are known as automorphic. Every algebraic curve of genus $ g > 1 $ is uniformized by automorphic functions (cf. Uniformization). The problem of the classification of elliptic curves also led to a study of the quotient $ D/G $ , but this situation is substantially different from the one just discussed. First, the group $ G $ has fixed points in $ D $ ; secondly, the manifold $ D/G $ is non-compact, though containing a finite Lobachevskii plane. The study of the general case of such groups, and of the respective quotients, plays an important role in modern arithmetical research.

If an algebraic curve $ X $ is defined over a non-closed field $ k $ , one of the most important problems is that of the existence and location of the set of rational points $ X(k) $ of $ X $ . In the case of a smooth projective curve over a finite field $ k $ a proof has been given of the inequality $ | N - q - 1 | \leq 2g \sqrt q $ , where $ N $ is the number of points on $ X $ that are rational over a finite extension $ L $ of $ k $ , $ q $ is the number of elements of $ L $ and $ g $ is the genus of $ X $ . This inequality is equivalent to the Riemann hypothesis concerning the zeros of the $ \zeta $ - function of $ X $ , viz. that all zeros of the $ \zeta $ - function lie on the vertical line $ \sigma = 1/2 $ ( cf. Zeta-function in algebraic geometry).

Now let $ X $ be an algebraic curve defined over the field of rational numbers $ \mathbf Q $ . For curves of genus zero the points of $ X ( \mathbf Q ) $ are relatively easily found, for elliptic curves the rational points constitute a finitely-generated group (if $ X ( \mathbf Q ) $ is non-empty), while for curves of genus $ g \geq 2 $ there is the Mordell conjecture to the effect that $ X ( \mathbf Q ) $ is finite.

If the ground field $ k $ is the field of rational functions $ k _{0} (B) $ of a smooth projective curve $ B $ , each smooth projective curve $ X $ over $ k $ is isomorphic to the general fibre $ X _ \eta $ of the morphism $ f: \ V \rightarrow B $ of a smooth projective algebraic surface $ V $ over $ k _{0} $ . This morphism is uniquely defined if it is assumed that its fibres do not contain exclusively curves of genus 1. The set of rational points $ X(k) $ is in one-to-one correspondence with the set of sections $ V(B) $ of $ f $ , and $ X(k) $ is finite for curves of genus $ g > 2 $ . Curves of genus 0 and 1 over the field $ k _{0} (B) $ are studied in the theory of algebraic surfaces (cf. Elliptic surface; Ruled surface).

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2]	R.J. Walker, "Algebraic curves" , Springer (1978) MR0513824 Zbl 0399.14016
[3]	D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701
[4]	C. Chevalley, "Introduction to the theory of algebraic functions of one variable" , Amer. Math. Soc. (1951) MR0042164 Zbl 0045.32301
[5]	J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) MR0103191
[6]	N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)
[7]	G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 MR0092855 Zbl 0078.06602
[8]	I.V. Dolgachev, V.A. Iskovskikh, "Geometry of algebraic varieties" J. Soviet Math. , 5 : 6 (1976) pp. 803–864 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 12 (1974) pp. 77–170

Comments

Estimate (1) above is due to G. Castelnuovo [a1]. A proof can also be found in [a2], [a3], [a4]. Reference [a2] also contains new results on the Riemann–Noether–Brill theorem, e.g. it has been proven that equality holds in the theorem for a generic curve in the sense of moduli; this reference also gives a survey of recent developments in the theory of algebraic curves.

The moduli space of curves $ {\mathcal M} _{g} $ is of general type, thus not unirational, if $ g $ is odd and $ g \geq 25 $ ( J.E. Harris and D. Mumford) or if $ g $ is even and $ g \geq 40 $ ( Harris), [a5].

The Mordell conjecture, i.e. every curve of genus at least two over a number field has only a finite number of rational points, has been proved by G. Faltings [a6].

References

[a1]	G. Castelnuovo, "Studies on the geometry of algebraic curves" Atti R. Acad. Sci. Torin , 24 (1889) pp. 196–223 (In Italian)
[a2]	E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1985) MR0770932 Zbl 0559.14017
[a3]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[a4]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 MR0463157 Zbl 0367.14001
[a5]	D. Mumford, J. Harris, "On the Kodaira dimension of the moduli space of curves" Invent. Math. , 67 (1982) pp. 23–88 MR0664324 Zbl 0506.14016
[a6]	G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. , 73 (1983) pp. 349–366 (Erratum: Invent. Math. 75(1984), 381) MR0718935 MR0732554 Zbl 0588.14026