Chasles-Cayley-Brill formula
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
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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
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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
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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
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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
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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
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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
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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
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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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This proves the birational invariance of and the Riemann–Hurwitz formula. For details, see [a2] and [a4].
References
[a1] | S.S. Abhyankar, "What is the difference between a parabola and a hyperbola" Math. Intelligencer , 10 (1988) pp. 36–43 |
[a2] | S.S. Abhyankar, "Algebraic geometry for scientists and engineers" , Amer. Math. Soc. (1990) |
[a3] | S.S. Abhyankar, "Field extensions" G.A. Pilz (ed.) A.V. Mikhalev (ed.) , Handbook of the Heart of Algebra , Kluwer Acad. Publ. (to appear) |
[a4] | C. Chevalley, "Introduction to the theory of algebraic functions of one variable" , Math. Surveys , 6 , Amer. Math. Soc. (1951) |
[a5] | J.L. Coolidge, "A treatise on algebraic plane curves" , Clarendon Press (1931) |
[a6] | R. Dedekind, H. Weber, "Theorie der algebraischen Functionen einer Veränderlichen" Crelle J. , 92 (1882) pp. 181–290 |
[a7] | S. Lefschetz, "Algebraic geometry" , Princeton Univ. Press (1953) |
[a8] | F. Severi, "Vorlesungen über algebraische Geometrie" , Teubner (1921) |
Chasles-Cayley-Brill formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chasles-Cayley-Brill_formula&oldid=22285