A note on the homology of ${\mathrm{\Sigma }}_n$, the Schwartz genus, and solving polynomial equations

We calculate a certain homological obstruction introduced by De Concini, Procesi and Salvetti in their study of the Schwartz genus of the fibration from the space of ordered configuration of points in the plane to the space of the unordered configurations. We show that their obstruction group vanishes in almost all, but not all, the hitherto unknown cases. It follows that if $n$ is not a power of a prime, or twice the power of a prime, then the genus is less than $n$. The case of $n=2p^k$ where $p$ is an odd prime remains undecided for some $p$ and $k$.