Projective geometry over a finite field

Example

Let $q$ be a prime power, let $\F_q$ be the finite field with $q$ elements, and let $n \ge 1$. The projective space $PG(n, q)$ is the projective space of $V = \F_q^{n+1}$. Its points are the $1$-dimensional subspaces of $V$, and its lines are the $2$-dimensional subspaces. In general its $k$-dimensional subspaces are the $(k+1)$-dimensional subspaces of $V$, with incidence given by inclusion.

Count of points: $V$ has $q^{n+1} - 1$ nonzero vectors, and each $1$-dimensional subspace contains $q - 1$ of them, so $PG(n, q)$ has $$\frac{q^{n+1} - 1}{q - 1} = q^n + q^{n-1} + \cdots + q + 1$$ points. Each line has $(q^2 - 1)/(q - 1) = q + 1$ points.

Check for $PG(3, 2)$: it has $15$ points and each line has $3$ points. Two distinct points span a unique line, so the number of lines is $\binom{15}{2} / \binom{3}{2} = 105 / 3 = 35$. Its planes are the $3$-dimensional subspaces of $\F_2^4$, and there are $15$ of them, one for each nonzero linear form up to scaling.

For $n = 2$, the plane $PG(2, q)$ is a finite projective plane of order $q$, and $PG(2, 2)$ is the Fano plane.