The Gauss--skizze decomposition is a Goresky-MacPherson stratification

We consider a new stratification of the space of configurations of $n$ marked points on the complex plane. Recall that this space can be differently interpreted as the space ${}^{\mathrm{D}}{\mathrm{P}\mathrm{o}\mathrm{l}}_n$ of degree $n>1$ complex, monic polynomials with distinct roots, the sum of which is 0. A stratum $A_{\sigma }$ is the set of polynomials having $P^{-1}(\mathbb{R}\cup \mathrm{ı}\mathbb{R})$ in the same isotopy class, relative to their asymptotic directions. We show that this stratification is a Goresky–MacPherson stratification and that from thickening strata a good cover in the sense of \v{C}ech can be constructed, allowing an explicit computation of the cohomology groups of this space.