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- W2895570302 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." @default.
- W2895570302 created "2018-10-12" @default.
- W2895570302 creator A5015512197 @default.
- W2895570302 date "2018-10-03" @default.
- W2895570302 modified "2023-09-24" @default.
- W2895570302 title "Cohomology of the space of polynomial maps on $mathbb{A}^1$ with prescribed ramification" @default.
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- W2895570302 doi "https://doi.org/10.1016/106881" @default.
- W2895570302 hasPublicationYear "2018" @default.
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