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

TL;DR

This paper computes the rational cohomology of moduli spaces of polynomial maps with prescribed ramification, revealing stability properties over complex numbers and differences in positive characteristic.

Contribution

It introduces a sheaf-theoretic approach to compute cohomology of ramification-prescribed polynomial map spaces, establishing rational stability and analyzing positive characteristic cases.

Findings

Rational cohomology is independent of degree n, indicating stability.

Computed étale cohomology in characteristic zero for these moduli spaces.

Found that étale cohomology does not stabilize in positive characteristic.

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.