Configurations of noncollinear points in the projective plane

Abstract

We consider the space $F_n$ of configurations of $n$ points in ${\mathbb{P}}^2$ satisfying the condition that no three of the points lie on a line. For $n=4,5,6$, we compute $H^{\ast }\left(F_n;\mathbb{Q}\right)$ as an ${\mathfrak{𝔖}}_n$–representation. The cases $n=5,6$ are computed via the Grothendieck–Lefschetz trace formula in étale cohomology and certain “twisted” point counts for analogous spaces over ${\mathbb{𝔽}}_q$.