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Abstract

This work discusses the topology of configurations of noncollinear points in the projective plane. As with classical configuration spaces, we can view configurations of noncollinear points in several distinct but compatible ways by leveraging their nature as varieties defined over the integers. For such varieties, a suite of comparison theorems provide a bridge between the singular cohomology and the \'{e}tale cohomology. Further, the Grothendieck--Lefschetz trace formula establishes a relationship between the \'{e}tale cohomology and the number of points on the variety over a finite field. In general, this relationship can be mixed by the action of the Frobenius automorphism, but when this action is known to be sufficiently tame, the original topological question can be reduced to a combinatorial problem of counting points. This work consists primarily of three components. We first extract sufficient topological structure to establish control of the Frobenius automorphism, showing that - at least up to configurations of six points - the topology may be understood in terms of a projective linear group and complements of hyperplane arrangements. We then elaborate on the reduction via the Grothendieck--Lefschetz trace formula to a set of point counting problems. We conclude by enumerating through the solutions to these point counting problems.

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