We are also interested in the null sets of systems of polynomials, and these are also considered algebraic varieties. Geometrically, this corresponds to studying the intersections of algebraic varieties.
Like working over the complex plane allows us to be more general when dealing with the shapes and dimensions of algebraic varieties themselves -- because every polynomial has a root over the complex numbers, etc. -- working with the projective complex plane allows us to be general when dealing with intersections of algebraic varieties -- because intersections work in a nice and general way on the projective complex plane.
The idea is that we want to be able to say that the intersection of two 1-dimensional varieties is "pretty much always" 0-dimensional, the intersection of two 2-dimensional varieties is "pretty much always" 1 -dimensional, etc. Here are the ways that this can go wrong, or we can have a "non-general" situation:
- The varieties are asymptotically parallel (e.g. two parallel lines/planes)
- The intersection has multiplicity (e.g. a paraboloid tangent to a plane)
- The varieties coincide on a region/have a common segment (e.g. two coincidental lines)
The first problem in particular is addressed by the projective complex numbers.
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| Parabola (red) and cubic curve (blue) in projective space |
The idea behind projective geometry is that any two parallel lines are said to meet at a particular "point at infinity" (and the point is the same in both directions, because we want to preserve the axiom that any two lines intersect at a single point). We thus want to add a point at infinity for each "direction" in the plane.
The standard construction to enable this is to define the projective complex set $P\mathbb{C}^n$ as the set of lines through the origin in $\mathbb{C}^{n+1}$, identifying each line with the point it intersects an off-origin copy of $\mathbb{C}^n$.
This construction is directly motivated by perspective drawings -- the origin from which the rays are extended is equivalent to the "eye" from which the perspective is taken (and the rays correspond to the actual rays of light through which the landscape is viewed), and the points at the horizon are seen by rays parallel to the ground.
Co-ordinates in the space $\mathbb{C}^{n+1}$ are then called homogenous co-ordinates for $P\mathbb{C}^n)$ -- and each point in $P\mathbb{C}^n$ can be represented by the co-ordinates of any point on the line in $\mathbb{C}^{n+1}$ it is identified with.
The definition of geometry is that it is the set of symmetries of a set. For the ordinary geometry we deal with, the manifold is $K^n$ (for some field $K$) and the symmetries are the rigid transformations (combinations of rotations, reflections and translations). In projective geometry, we deal with the symmetries of the manifold $PK^n$ under the so-called projective transformations (or homographies), which generalize affine transformations in that they cannot be written as matrices.
The key idea behind projective transformations is that the points at infinity are not truly "special" or "different" from ordinary points, so transformations that transform between Euclidean points and points at infinity are symmetries of the projective space.
The obvious way to do this is to define projective transformations on $PK^n$ as restrictions of (non-singular) linear transformations in homogenous co-ordinates (i.e. on $K^{n+1}$), because these are precisely the transformations that send lines to lines (which are identified with points of $PK^n$).
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| The transformation sending the green line to the lime line is a homography on the projective space. |
To express a homography within the projective space, we just evaluate $\phi A\phi^{-1}$, where $\phi$ is the conversion to homogenous co-ordinates and $A$ is the linear transformation in $K^{n+1}$.
Exercise: show that the homographies on the projective line $PK^1$ are precisely the functions of the form $f(z)=\frac{az+b}{cz+d}$.
Exercise (homogenous polynomials): show that a polynomial in projective space is a homogenous polynomial (all its component monomials have the same total degree) in homogenous co-ordinates.
E.g. the polynomial $x^2-y=0$ can be written in homogenous co-ordinates as $x_0^2-x_1x_2=0$.
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