# Configurations of noncollinear points in the projective plane

## TL;DR

This paper studies the topological and algebraic structure of configuration spaces of n points in the projective plane with no three collinear, computing their rational cohomology for small n using advanced algebraic geometry techniques.

## Contribution

It explicitly computes the rational cohomology of these configuration spaces for n=4, 5, 6, revealing their structure as symmetric group representations.

## Key findings

- Computed H^*(F_n; Q) for n=4, 5, 6 as S_n-representations
- Used Grothendieck--Lefschetz trace formula for n=5, 6
- Applied point counting over finite fields for cohomology calculations

## Abstract

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 $n = 4, 5, 6$, we compute $H^*(F_n; \mathbb{Q})$ as an $S_n$-representation. The cases $n = 5, 6$ are computed via the Grothendieck--Lefschetz trace formula in \'etale cohomology and certain "twisted" point counts for analogous spaces over $\mathbb{F}_q$.