## Projective anabelian curves in positive characteristic and descent theory for log-étale covers |

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### Contents

Topological Coincidence of Maps | 3 |

The Van Kampen Theorem | 9 |

Logarithmic Geometry | 25 |

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### Common terms and phrases

2-complex affine algebraically closed field Anabelian Geometry automorphism base change base extension bijection canonical center free closed point cohomology Corollary curve of genus defined denote descent theory diagram discrete valuation ring double points etale locally F-cohomologically rigid finite fields Frobenius module fs monoids fs-log schemes fs-log structure G-torsors Galois category Galois group genus g geometric point geometrically connected varieties graph Grothendieck group-theoretical Hence henselian homeomorphism hyperbolic curves induced injective irreducible components isomorphism isotrivial Kampen Theorem ket cover Kummer Kurven Lemma Let f Let X/S level structure log blow-up log scheme log structure log-etale log-radicial log-smooth monoids monomorphism morphism natural map noetherian normal subgroup prime number pro-finite group Proposition pullback quasi-compact quotient reduction resp Revlog semistable curve sheaf sheaves short exact sequence space Spec(Z[P special fibre specialisation map strict henselian surjective topological torsion torsors universal homeomorphism vanishes yields