Let be an irreducible algebraic plane curve of degree , given by an equation where is an irreducible bivariate polynomial of degree over a ground field (cf. also Algebraic curve). For simplicity is assumed to be algebraically closed (cf. also Algebraically closed field), although most of what is said below can be suitably generalized without that assumption. For the basic field theory involved, see [a6] (or the modernized version [a4]) and [a3]. For much of the geometry to be discussed, see [a5] and [a8]; in particular, for the idea of points at infinity of , see [a1]. For an interplay between the geometry and the algebra, see [a2].
One starts by analyzing when the curve can be rationally parametrized. For example, the unit circle has the rational parametrization
Likewise, the cuspidal cubic has the rational parametrization and . However, the non-singular cubic does not have any rational parametrization. To obtain the parametrization of the circle, one cuts it by a line of slope through the point and notes that it meets the circle in the variable point
For the cuspidal cubic one takes a line through the cusp and notes that it meets the cubic in the variable point . This works because a line meets a circle in points, and it meets a cubic in points. In case of a cuspidal cubic two intersections are absorbed in the cusp. In case of a non-singular cubic there is no such point for the absorption. Generalizing this one can show that the curve cannot have more than double points and if it does have that many, then it can be parametrized rationally. To this end one first notes that a bivariate polynomial of degree has
coefficients and hence the dimension of the system of curves of degree passing through double points of is . Next, by the Bezout theorem (which is the oldest theorem in algebraic geometry), a curve of degree and a curve of degree , having no common component, meet in points, counted properly. In the proper counting a double point of should be counted twice. Thus, the number of free points in which meets a curve of degree passing through the double points of is . Since is also the dimension of the system , the members of which pass through fixed simple points of form a pencil, i.e., a one-parameter family , a variable member of which meets in one variable point whose coordinates are single-valued, and hence rational, functions of . If had an extra double point, then one can take a value of so that goes through it and this would make the properly counted intersections of and to be , contradicting the Bezout theorem because is irreducible and has smaller degree.
Provisionally defining the genus of (cf. also Genus of a curve) by
one always has , and is rational, i.e., has a rational parametrization. To make the reverse implication also true, one must learn to count the double points properly. To begin with, one must include singularities of at infinity. Next, by looking at the curve , where with , which is obviously rational and has a -fold point at the origin and an -fold point at infinity as its only singularities, one decides to count a -fold point as double points. Before discussing infinitely near singularities, one notes that the degree of can be geometrically characterized as the number of points in which a general line meets it.
Likewise, the multiplicity of a point of can be characterized geometrically by saying that is equal to the number of points in which a generic line through meets outside ; is a simple or singular point of according as or . Algebraically, by translating the coordinates one may assume to be the origin , and then is the order of , i.e., has terms of degree but none of degree . By making the quadratic transformation and one gets , where : is the proper transform of . The exceptional line meets in points whose multiplicities add up to . These are the points of in the first neighbourhood of . Points in the first neighbourhoods of these points are the points of in the second neighbourhood of , and so on. It is easily seen that all points in a high enough neighbourhood of are simple. Now is counted as double points, where
with the summation extended over all points in the various neighbourhoods of , including ; here is the multiplicity of ; clearly: is a simple point of . One arrives at the exact genus formula
with summation over all points of . One always has ; and is rational.
It turns out that is a birational invariant of , i.e., it remains unchanged when undergoes a birational transformation (cf. also Birational morphism). The residue class ring of the polynomial ring modulo the ideal generated by is the affine coordinate ring of and is denoted by . Note that where , are the images of , in . The quotient field of is the function field of . A birational correspondence between curves and is an almost one-to-one correspondence; it is given by a -isomorphism between and . So one should be able to define directly in terms of . Following C.G.J. Jacobi one takes any differential of (cf. also Differential field), i.e., an expression of type with , and shows that if the differential is not zero, then the number of its zeros minus the number of its poles equals . Having brought the point of to the origin, its local ring is defined to be the subring of consisting of all quotients where , are polynomials with (cf. also Local ring); its unique maximal ideal consists of the above quotients with . Let be the conductor of , i.e., the largest ideal in which remains an ideal in the integral closure of in . It can be shown that is the length of in , i.e., the maximal length of strictly increasing chains of ideals in ; moreover, is the length of in , which is a ubiquitous result having two dozen proofs in the literature. The ring has a finite number of maximal ideals and localizing at them gives discrete valuation rings; as varies over all points of , including those at infinity, these discrete valuation rings vary over the Riemann surface of , i.e., the set of all discrete valuation rings whose quotient field is and which contain . Let denote the localizations of at the various maximal ideals in (cf. also Localization in a commutative algebra); one calls the centre on of the members of ; note that is a simple point of , and hence for all except a finite number of points of , the set has exactly one member. For any and non-zero one puts
with and ; take with and define
one calls a uniformizing parameter of . Now the number of zeros minus number of poles of equals taken over all in . For any point of , not at infinity, one has Dedekind's formula
where is the different ideal in defined by saying that with for every .
For , let be an irreducible algebraic plane curve such that is a finite separable algebraic field extension of of field degree (cf. also Extension of a field; Separable extension). This defines a correspondence between and , and hence between and ; namely, and correspond if and only if for some one has and . Let be the genus of , let the different be the integer-valued function on whose value at in is given by , where is a uniformizing parameter of , and let with summation over all . Then the Riemann–Hurwitz formula says that
and this gives rise to the Zeuthen formula
Now suppose there is a -isomorphism . Then is called a fixed place of the correspondence if . The Chasles–Cayley–Brill formula says that under suitable conditions, the number of these, counted properly, equals , where the integer is called the valence of the correspondence. For details see [a7], pp. 189–194.
In case is the field of complex numbers, to describe Riemann's approach one topologizes to make it into a compact orientable two-dimensional real manifold, and hence into a sphere with handles (cf. also Riemann surface). Likewise, is made into a sphere with handles. Triangulate by including all the branch points as vertices, and lift this triangulation to a triangulation of . Let and be the vertices, edges, faces of the bottom and top triangulations respectively. Then , , , and hence by the Euler–Poincaré theorem one obtains
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Chasles-Cayley-Brill formula. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Chasles-Cayley-Brill_formula&oldid=23779