1 Projective space

The ordinary plane R^{2} has the problem that parallel lines do not meet. So, introduce ‘ideal’ points at infinity, and an ideal line containing those points, to get a projective plane in which two points always determine a unique line, and two lines always determine a unique point.

A model of this projective plane is found in R^{3} by taking the lines through the origin as projective points. The original plane can be taken to be the plane Z = 1. Each line ⟨(x, y, z)⟩ with z ≠ 0 hits this plane in a unique point, namely (x/z, y/z, 1), the ‘old point’ (x/z, y/z). The lines ⟨(x, y, 0)⟩ correspond to the ideal points, the plane Z = 0 to the ideal line (the ‘line at infinity’).

Now generalize to arbitrary fields and dimensions.

Let V be a vector space over a field F. The associated projective space P V is the lattice of subspaces of V (where incidence is inclusion). Projective points, lines, planes, ..., i-spaces, ... are d-dimensional linear subspaces of V with d = 1, 2, 3, ..., i + 1, ... . We shall always use vector space dimension, never projective dimension. Points will be projective points - elements of V are called vectors.

When V is coordinatized, a projective point is a 1-space {(λx_{1}, ..., λx_{n})|λ ∈ F } and we can say that this point has coordinates (x_{1}, ..., x_{n}) provided we agree that (x_{1}, ..., x_{n}) and (αx_{1}, ..., αx_{n}) denote the same point (for α ∈ F, α ≠ 0). Such coordinates are called homogeneous coordinates.

We would like to describe sets of points by an equation f(X_{1}, ..., X_{n}) = 0. For this to be meaningful, one wishes f(x_{1}, ..., x_{n}) = 0 iff f(αx_{1}, ..., αx_{n}) = 0 (for α ∈ F, α ≠ 0). This leads us to consider homogeneous polynomials, that is, polynomials in which the total degree of each term is the same (say d). These satisfy our restraint, since f(αx_{1}, ..., αx_{n}) = α^{d}f(x_{1}, ..., x_{n}).

2 Chevalley-Warning

Below we shall need that a quadric on a projective plane over a finite field F has at least one point. This is a very special case of the following.

Theorem 2.1 Given homogeneous polynomials f_{1}, ..., f_{m} over F (of respective degrees d_{1}, ..., d_{m}) in the variables X_{1}, ..., X_{n}, where ∑ d_{i} < n. Then there is a common zero (x_{1}, ..., x_{n}) ≠ (0, ..., 0).

Proof: Consider

∑_{x∈F^{n}} ∏_{i}(1 − f_{i}(x)^{q−1}).

It is the total number of solutions (considered as an element of F) of f_{1} = ... = f_{m} = 0. But ∑_{z∈F} z^{j} = 0 unless j is a nonzero multiple of q − 1, so in the expansion of the above sum only terms of degree at least n(q − 1) contribute, but there are no such terms. Consequently, the number of solutions is 0 (mod charF). Since (0, ..., 0) is a solution, there are also other solutions. □

It follows that a quadric in dimension at least 3 is nonempty.

3 Quadrics over finite fields

A homogeneous polynomial of degree one (a linear form) defines a hyperplane. A homogeneous polynomial of degree two (a quadratic form) defines a quadric. We want to classify quadrics in projective spaces over finite fields.

3.1 Quadratic forms

Let us first describe quadratic forms in a coordinate-free way. A quadratic form is a map Q : V → F such that Q(λu) = λ^{2}Q(u) (for λ ∈ F, u ∈ V) and having the property that B defined by B(u, v) = Q(u + v) − Q(u) − Q(v) is a bilinear form. We have B(u, u) = 2Q(u), so if charF ≠ 2 then we can retrieve Q from B. (But if charF = 2, then many quadratic forms yield the same bilinear form.)

3.2 Nondegeneracy

Two points ⟨u⟩ and ⟨v⟩ are called orthogonal if B(u, v) = 0. If U is a subspace of V, then U^{⊥} denotes the subspace of V consisting of the vectors orthogonal to each element of U. The bilinear form B is called nondegenerate if V^{⊥} = 0. The quadratic form Q is called nondegenerate if Q does not vanish on any point in V^{⊥}.

3.3 Orthogonal direct sums

We write V = U ⊥ W if V = U ⊕ W (that is, V = U + W and U ∩ W = 0) and B(u, w) = 0 for all u ∈ U, w ∈ W. If this is the case, then Q is determined by its restrictions to U and W. Indeed, Q(u + w) = Q(u) + Q(w). And conversely, given arbitrary quadratic forms Q_{U} on U and Q_{W} on W, the formula Q(u + w) = Q_{U}(u) + Q_{W} (w) defines a quadratic form Q on U ⊕ W, and for this form V = U ⊥ W.

3.4 Classification

Thus, it suffices to classify pairs (V, Q) where V cannot be decomposed as orthogonal direct sum. In particular, we may suppose that V^{⊥} = 0 (unless V = V^{⊥} is of dimension 1 and we have Q = 0).

Now suppose dimV ≥ 3. Then by Chevalley-Warning there is a point ⟨u⟩ with Q(u) = 0. Since V^{⊥} = 0 there is a vector w with B(u, w) = λ ≠ 0. With v = w + βu we find Q(v) = Q(w) + βλ, so for β = −Q(w)/λ we have Q(v) = 0. After scaling v we may assume that B(u, v) = 1. Thus, we have found a line H = ⟨u⟩+⟨v⟩ with quadratic form defined by Q(u) = Q(v) = 0 and B(u, v) = 1. (With u and v as unit vectors, the form is X_{1}X_{2}.) Such a line is called a hyperbolic line. Clearly, H is nondegenerate, and we have V = H ⊥ H^{⊥} where also H^{⊥} is nondegenerate. In this way we can peel off hyperbolic lines until the dimension n of V has become less than 3.

If n = 0 then P V has no points and Q = 0.

If n = 1 then P V has a single point and the number of nonisomorphic possibilities equals 1+|F^{∗}/F^{∗2}|. Indeed, if the single point is ⟨u⟩, then changing u by a constant α changes Q(u) by α^{2}, so either Q(u) = 0 (u is singular) or Q(u) lies in one of the cosets of F^{∗2} in F^{∗}. For finite fields this means that either charF = 2 and there are just two possibilities: Q(u) vanishes or not, or charF is odd, and there are three possibilities: Q(u) is zero, a square or a nonsquare.

If n = 2 and V is not a hyperbolic line, then Q never vanishes on P V, and the line is called an elliptic line.

Thus: V is the orthogonal direct sum of its radical V^{⊥}, a number of hyperbolic lines, and perhaps a single point or an elliptic line.

By change of coordinates H ⊥ P can be transformed into H ⊥ P′ where Q(P) consists of squares, and Q(P′) of nonsquares. So, for odd n > 1 the type of the point occurring in the decomposition on a nondegenerate quadratic form is irrelevant. Also, any two elliptic lines are equivalent. The parity of the number of elliptic lines occurring in a decomposition is an invariant (found from the determinant of Q).

Thus: over a finite field we have in even dimension n > 0 precisely two types of nondegenerate quadrics: hyperbolic quadrics that are an orthogonal direct sum of hyperbolic lines, and elliptic quadrics, that are not. In odd dimension n > 1 there is only one type of nondegenerate quadric.

The maximal totally singular subspaces (subspaces where Q vanishes identically) have dimension m for a hyperbolic quadric in n = 2m dimensions, and m − 1 for an elliptic quadric in n = 2m dimensions, and m for a nondegenerate quadric in n = 2m + 1 dimensions.

The (vector space) dimension of the maximal totally singular subspaces is called the Witt index.