Cohomology of the space of polynomial maps on ${\mathbb{A}}^1$ with prescribed ramification

Abstract

In this paper we study the moduli spaces $Sim{p}_n^m$ of degree $n+1$ morphisms ${\mathbb{A}}_K^1\to {\mathbb{A}}_K^1$ with "ramification length $<m$" over an algebraically closed field $K$. For each $m$, the moduli space $Sim{p}_n^m$ is a Zariski open subset of the space of degree $n+1$ polynomials over $K$ up to $Aut({\mathbb{A}}_K^1)$. It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes -- here we are prescribing, instead, the ramification data. Exploiting the topological properties of the poset that encodes the ramification behaviour, we use a sheaf-theoretic argument to compute $H^{\ast }(Sim{p}_n^m(\mathbb{C});\mathbb{Q})$ as well as the étale cohomology $H_{et}^{\ast }({Sim{p}_n^m}_{/K};{\mathbb{Q}}_{\ell })$ for $charK=0$ or $charK>n+1$. As a by-product we obtain that $H^{\ast }(Sim{p}_n^m(\mathbb{C});\mathbb{Q})$ is independent of $n$, thus implying rational cohomological stability. When $charK>0$ our methods compute $H_{et}^{\ast }(Sim{p}_n^m;{\mathbb{Q}}_{\ell })$ provided $charK>n+1$ and show that the étale cohomology groups in positive characteristics do not stabilize.

28 pages, 7 figures, few minor corrections