On Cohomology of The Space of Linear Generic Points in Three Dimensional Projective Space

In this paper, we first introduce a type of generalized configuration spaces $C_n^m$ and $\p C_n^m$, namely, it consists the $n$-tuples in affine or projective $m$ spaces where any $m+1$ points are not contained in a hyperplane. The spaces can be considered as complements of affine and projective algebraic sets respectively and hence are quasi-projective varieties. There is a free action of $AGL_{m+1}(\C)$ and $PGL_{m+1}(\C)$ on $C_n^m$ and $\p C_n^m$ respectively, which gives the spaces a structure as trivial bundles. Also, similar to the classic configuration spaces, there is a natural $S_n$ action on these spaces and that puts an induced $S_n$-represnetation structure on the cohomology.Based on these structures, there are a few result in fundamental group and cohomology group of these spaces studied by other mathematicians. We will introduce some of the results and focus on the cohomology of $\p C_n^m$. To go one more step further, let $X_n^m$ be the quotient of $\p C_n^m$ by $PGL_{m+1}(\C)$, we will study the cohomology of $X_n^3$ as a $S_n$ representation for $n=6$ and $7$. In the case that $n=7$, our result is based on the assumption that the cohomology groups $H^k(X_n^m)$ are pure of Hodge-Tate type $(k,k)$. The calculation is done by the technique of twisted point counting introduced by \cite{Church_2014}. To utilize the point counting technique, we will discuss the background that makes it possible and introduce the Grothendiecck-Lefchetz formula which is the main foundation of the calculation.