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

Abstract

In this paper we study the moduli spaces $Simp^m_n$ of degree $n+1$ morphisms $ \mathbb{A}^1_{K} \to \mathbb{A}^1_{K}$ with "ramification length $<m$" over an algebraically closed field $K$. For each $m$, the moduli space $Simp^m_n$ is a Zariski open subset of the space of degree $n+1$ polynomials over $K$ up to $Aut (\mathbb{A}^1_{K})$. 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 $H^*(Simp^m_n(\mathbb{C}); \mathbb{Q})$ as well as the \'etale cohomology $H^*_{et}({Simp^m_n}_{/K}; \mathbb{Q}_{\ell})$ for $char K=0$ or $char K> n+1$. As a by-product we obtain that $H^*(Simp^m_n(\mathbb{C}); \mathbb{Q})$ is independent of $n$, thus implying rational cohomological stability. When $char K>0$ our methods compute $H^*_{et}(Simp^m_n; \mathbb{Q}_{\ell})$ provided $char K>n+1$ and show that the \'etale cohomology groups in positive characteristics do not stabilize.