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Reduction and lifting of special metacyclic covers. (English) Zbl 1042.14005

The author sets the scene for his paper in the context of a paper of M. Raynaud [ibid. 32, 87–126 (1999; Zbl 0999.14004)], who gave a criterion for good reduction of Galois covers of the projective line that are ramified at three points. He introduced the notion of an auxiliary cover, which enables one to analyze Galois covers with bad reduction in characteristic \(p\), in terms of subgroups \(H\) with the same type of bad reduction, in which \(H\) is a solvable quotient of a subgroup of \(G\). For example if \(p\) is a proper divisor of the order of \(G\), then \(H\) is a metacyclic group, isomorphic to \(\mathbb{Z}/p\times \mathbb{Z}/m\). Thus the study of bad reduction can be reduced to the study of covers with certain solvable (in the case under discussion, metacyclic) Galois groups.
The foregoing is a motivation for the study of metacyclic covers of \(\mathbb{P}_1\) with Galois group isomorphic to \(\mathbb{Z}/p\times \mathbb{Z}/m\), which arise as auxiliary covers of \(\mathbb{P}^1\), having three branch points and prime-to-\(p\) ramification.
The paper begins with a characterisation of metacyclic covers, \(f: Y\to \mathbb{P}^1\) which are the composition of a ramified \(m\)-cyclic cover \(Z\to \mathbb{P}^1\) and an étale \(p\)-cyclic cover \(Y\to Z\), with Galois group \(\mathbb{Z}/p\times\mathbb{Z}/m\). The theory of such special metacyclic covers is developed very clearly and geometrical examples are given, concluding with a connection between the existence of étale Galois covers of the affine line in characteristic \(p\), related to Abhyankar’s conjecture.
The author then investigates the problem of the reduction of a special cover to characteristic \(p\) which is determined by what he calls special degeneration data, which are given essentially by an \(m\)-cyclic cover \(\overline Z_0\to \mathbb{P}^1_k\) of the projective line in characteristic \(p\), together with a logarithmic differential form \(\omega_0\) on \(Z_0\), where \(Z_0\) is defined in the proof of the author’s Theorem A and, together with \(\omega_0\), determines a pair \((Z_0,\omega_0)\) where \(\overline Z_0\to \mathbb{P}^1_k\) is an \(m\)-cyclic cover branched at certain points \(\tau_1,\dots, \tau_n\).
Finally, the author shows that any special degeneration datum can be lifted to a special cover.
The geometrical motivation for the study of Galois covers is clearly presented, and the author promises further developments in subsequent papers.

MSC:

14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
14H30 Coverings of curves, fundamental group
14E20 Coverings in algebraic geometry

Citations:

Zbl 0999.14004
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References:

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