Moduli of certain wild covers of curves
Abstract
A fine moduli space is constructed, for cyclic-by-$\mathsf{p}$ covers of an affine curve over an algebraically closed field $k$ of characteristic $\mathsf{p}>0$. An intersection of finitely many fine moduli spaces for cyclic-by-$\mathsf{p}$ covers of affine curves gives a moduli space for $\mathsf{p}'$-by-$\mathsf{p}$ covers of an affine curve. A local moduli space is also constructed, for cyclic-by-$\mathsf{p}$ covers of $Spec(k((x)))$, which is the same as the global moduli space for cyclic-by-$\mathsf{p}$ covers of $\mathbb{P}^1-\{0\}$ tamely ramified over $\infty$ with the same Galois group. Then it is shown that a restriction morphism is finite with degrees on connected components $\mathsf{p}$ powers: There are finitely many deleted points of an affine curve from its smooth completion. A cyclic-by-$\mathsf{p}$ cover of an affine curve gives a product of local covers with the same Galois group of the punctured infinitesimal neighbourhoods of the deleted points. So there is a restriction morphism from the global moduli space to a product of local moduli spaces.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2019
- DOI:
- 10.48550/arXiv.1904.03763
- arXiv:
- arXiv:1904.03763
- Bibcode:
- 2019arXiv190403763Z
- Keywords:
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- Mathematics - Algebraic Geometry
- E-Print:
- some typos corrected and minor changes made