  Let $m_{12}$, $m_{13}$, ..., $m_{n-1,n}$ be the slopes of the $\binom{n}{2}$
lines connecting $n$ points in general position in the plane. The ideal $I_n$
of all algebraic relations among the $m_{ij}$ defines a configuration space
called the {\em slope variety of the complete graph}. We prove that $I_n$ is
reduced and Cohen-Macaulay, give an explicit Gr\"obner basis for it, and
compute its Hilbert series combinatorially. We proceed chiefly by studying the
associated Stanley-Reisner simplicial complex, which has an intricate recursive
structure. In addition, we are able to answer many questions about the geometry
of the slope variety by translating them into purely combinatorial problems
concerning enumeration of trees.
