  We describe sequences of blowups of $\overline{M}_{0,5} \times
\overline{M}_{0,5}$ and $\mathbf{P}^2 \times \mathbf{P}^2$ yielding a small
resolution of the stable pair compactification $\overline{M}(3,6)$ of the
moduli space $M(3,6)$ of six lines in $\mathbf{P}^2$. These blowup sequences
can be viewed, respectively, as generalizations of Keel's and Kapranov's
constructions of $\overline{M}_{0,n}$. We use these blowup sequences to
describe the intersection theory of $\overline{M}(3,6)$. In particular, we show
that the Chow ring of any small resolution of $\overline{M}(3,6)$ has a
presentation analogous to Keel's presentation of $A^*(\overline{M}_{0,n})$, and
the Chow ring of $\overline{M}(3,6)$ is an explicit subring of the Chow ring of
one of these small resolutions. We also introduce higher-dimensional versions
of the $\psi$-classes on $\overline{M}_{0,n}$, and describe their intersections
on $\overline{M}(3,6)$. Finally, we use our results to obtain an independent
proof of Luxton's result that $\overline{M}(3,6)$ is the log canonical
compactification of $M(3,6)$.
