Configurations of noncollinear points in the projective plane

We consider the space $F_n$ of configurations of $n$ points in $P^2$ satisfying the condition that no three of the points lie on a line. For $n = 4, 5, 6$,wecompute$H^*(F_n; $\mathbb{Q}$)$asan$S_n$-representation\mathrm{.}Thecases$n = 5, 6$ are computed via the Grothendieck–Lefschetz trace formula in 'etale cohomology and certain “twisted” point counts for analogous spaces over ${\mathbb{F}}_q$.