Today I feel like explaining a little bit of algebraic geometry, namely a theorem by Max Noether and Castelnuovo on the so-called Cremona group of the plane. (My intention is to explain what the theorem states, not to prove it, which would take far more space than an entry here would allow. I hope I can make what follows clear to people with a relatively modest knowledge of mathematics—perhaps an undergraduate level.) Incidentally, let me mention that Max Noether is the father of Emmy Noether, who was certainly the most remarkable woman mathematician ever (and far more important to mathematics than Max Noether).

We concern ourselves with birational transformations of
the plane, so let us first explain what this means. A rational
map of the plane to itself is given, by definition, by two
expressions, (`x`′,`y`′), giving the
coordinates of an image point in function of those, (`x`,
`y`), of the original point, in which we impose that
`x`′ and `y`′ be *rational
functions* of `x` and `y`, in other words
quotients of two polynomials in these two variables (which we can
assume, after elimination, to have no common factor), say, with real
coefficients, the denominator being—of course—not
identically zero. For example, letting `x`′=0 and
`y`′=0 maps the entire point to the origin (0,0);
letting `x`′=−`x` and
`y`′=`y` defines the symmetry with respect to
the vertical axis; letting `x`′=`x` and
`y`′=`x`² gives a rational map that projects
the entire plane onto the parabola with equation
`y`=`x`², and so on. Note that a rational map is
not always defined everywhere, since we have allowed
polynomials in the denominator: for example letting
`x`′=1/`x` and
`y`′=1/`y` gives a rational map that is
defined only so long as `x` and `y` are both nonzero
(i.e. away from the coordinate axes), but we still call this a
rational map of the plane to itself; since the denominator cannot be
identically zero, there are always some points in the plane (many
points, in fact: in a topological sense they are “dense”)
for which the mapping is defined. Now we can generally
compose two rational maps of the plane to itself, merely by
replacing the `x` and `y` variables in the
“outer” map by the `x`′ and
`y`′ given by the “inner” map; for example,
composing the former map (`x`′=1/`x` and
`y`′=1/`y`) with itself gives the identity map
(`x`′=`x` and
`y`′=`y`), simply because
1/(1/`t`)=`t`. (Note that we cannot *always*
compose two rational maps, because the composition might make sense
nowhere, in other words it might end up giving an identically zero
denominator, something we have excluded.)

When a rational map of the plane to itself is such that there
exists another map which, when composed with it, (makes sense and)
gives the identity map (viz. `x`′=`x` and
`y`′=`y`), we say that the map (either of the
two maps, actually) are birational transformations of the
plane. For example, our previous example
(`x`′=1/`x` and
`y`′=1/`y`) is a birational transformation of
the plane, since, as we have explained, when composed with itself, it
gives the identity map. On the other hand, the constant rational map
(`x`′=0 and `y`′=0) is *not* a
birational transformation of the plane, since composing it with
anything (in any order) gives a constant map if it makes sense at all.
In perhaps more intuitive terms, a birational transformation of the
plane is a rational map of the plane to itself which can be inverted
by another rational map. (It is not exactly a bijection in the
function sense, because it might not be defined everywhere. However,
it is “mostly” a bijection.) The misfortune of
composition not always being defined does not happen for birational
transformations: the composition of two birational transformations of
the plane always exists (makes sense) and is always itself a
birational transformation; and, of course, by definition, any
birational tarnsformation has an inverse which is also a birational
transformation. (For those who know what that means, birational
transformation of the plane form a group, the Cremona group
of the plane. Note that it does not suffice for a rational map to be
bijective in order for it to be a birational transformation: for
example, `x`′=`x`³ and
`y`′=`y`³ defines a rational map which is
bijective because cube roots always exist and are unique, but which is
not a birational transformation because cube roots are not rational
functions.)

Now let us consider some particular birational transformatins.
First of all, we have those of the following form: `x`′
=
(`a`_{1}·`x`+`b`_{1}·`y`+`c`_{1})
/
(`a`_{0}·`x`+`b`_{0}·`y`+`c`_{0})
and `y`′ =
(`a`_{2}·`x`+`b`_{2}·`y`+`c`_{2})
/
(`a`_{0}·`x`+`b`_{0}·`y`+`c`_{0}),
in which the `a`_{i},
`b`_{i} and
`c`_{i} are real numbers: in other words,
`x`′ and `y`′ can be expressed as
quotients of *first degree* (aka affine) polynomials, with
equal denominator; I'm not claiming that *any* such form
defines a birational transformation, but it turns out that almost all
do (all one needs to impose is the nonvanishing of the determinant of
the 3×3 matrix of coefficients, and then the inverse transformation is
given in the same form by the inverse matrix); those that *are*
of this form are called projective transformations of the
plane. For example, all plane symetries, rotations, translations, and
much else, are all projective transformations. The composition of two
projective transformations of the plane is a projective
transformation, and the inverse of a projective transformation is a
projective transformation (in technical terms, this means that
projective transformations form a subgroup of the Cremona group).
Basically, the projective transformations are the
“uninteresting” birational transformations: they take
lines into lines, so they are geometrically boring. (Another fact
worthy of note is that given four distinct points in the plane, no
three of which are aligned, and given another quadruplet of points
satisfying the same condition, there is a unique projective
transformation taking one set in the other in the prescribed
order.)

Now let us consider two very simple examples of birational
transformations which are *not* projective transformations: the
first one, which has `x`′=1/`x` and
`y`′=1/`y`, we have already mentioned; the
second one is `x`′ =
`x`/(`x`²+`y`²) and `y`′ =
`y`/(`x`²+`y`²). Both give the identity
when composed with themselves, as is easily checked, so both are,
indeed, birational transformations of the plane. The first one
corresponds to inverting the two (cartesian) coordinates
independently, and the second corresponds to inverting the radial
coordinate of polar coordinates (in other words, the distance to the
origin) while keeping the angular coordinate fixed. Call these two
transformations the basic Cremona transformations of the
plane.

Well, the remarkable fact, which is the statement of Max Noether's transformation theorem, is that these two basic Cremona transformations are essentially the “only” ones we can form, or, rather, they are the building blocks for all other birational transformations; the precise statement is: any birational transformation of the plane can be written as the composition of projective transformations and the two basic Cremona transformations which we have given (of course it may be necessary to use either of them multiple times in the composition). So any birational plane transformation, no matter how complex, can be reduced to these basic transformations. (In reality, the theorem is somewhat simpler if we work over the complex numbers instead of the reals as we have done: then we need only one basic Cremona transformation, and either one will do. Over the reals we need two. But I did not want to make this entry more complicated by considering the plane with complex coordinates.)

To give an idea of why this is remarkable, we can consider the
situation in other dimensions: is is straightforward to define
birational transformations of the line, of three-dimensional space,
and in fact, of `n`-dimensional space for any `n`,
and among these are the obviously defined projective transformations.
Now it turns out that in dimension one, the *only* birational
transformations are the projective transformations (also called
homographies): there are simply no others (and there is no
Cremona transformation). In dimension three or more, it is not
possible to find a finite number of particular birational
transformations which, when composed with the projective
transformations, will span all other birational transformations: the
Cremona group is simply too complicated, and it is hardly possible to
say anything useful about it.

Well, I don't know whether this is all very enlightening; at least, it is a little but elegant bit of algebraic geometry.