Can anyone please provide me with a reference on $H^n(M\_{0,n+3},{\mathbb C})$ where $M\_{0,n+3}$ is the (affine) scheme parametrizing $n+3$ labeled distinct points on ${\mathbb C\mathbb P}^1$? I am looking for combinatorial description, dimension, weight filtration, etc.



On $n$-th cohomology of $M_{0,n+3}$
$M\_{0,n+3}$ can be identified with the set $(z\_1, z\_2, \ldots, z\_n) \in \mathbb{C}^n$ so that $z\_i \neq 0$, $z\_i \neq 1$ and $z\_i \neq z\_j$ for every $i \neq j$. In other words, it is the complement of $2n+\binom{n}{2}$ hyperplanes in $\mathbb{C}^n$. The cohomology of the complement of a hyperplane arrangement was computed by Orlik and Solomon; [here](http://www.math.cornell.edu/~levine/orliksolomonfinal.pdf) is an expository paper which covers the full result in great detail. 


In summary, consider the complement in $\mathbb{C}^n$ of the hyperplanes given by linear equations $\lambda\_1(z)=0$, $\lambda\_2(z)=0$, ..., $\lambda\_N(z)=0$. The cohomology ring is generated in $H^1$ by the forms $d \lambda\_i/\lambda\_i$; these are Hodge-Tate of type $(1,1)$, so $H^n$ is entirely of type $(n,n)$. There are explicit combinatorial formulas for the dimensions in terms of matroid theory. In your case, I get that $\dim H^n(M\_{n+3}) = (n+1)!$. 


