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Cohomology of the space of polynomial maps on $\mathbb{A}^1$ with prescribed ramifications. (arXiv:1810.01934v1 [math.AG])
来源于:arXiv
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}_{\el 查看全文>>