Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves (original) (raw)
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Pro-l birational anabelian geometry over algebraically closed fields I
Eprint Arxiv Math 0307076, 2003
Zariski prime divisors of K|k, where K is any function field over an algebraic closure k of a finite field such that td(K|k) > 1. b) Part of the local theory is to show that in the context from above, the whole inertia structure which is significant for us is encoded in G ℓ K. c) Third, a more technical result by which we recover the "geometric sets of prime divisors" of function fields K|k as above. By definition, a geometric set of prime divisors of K is the set of Zariski prime divisors D X ⊂ D K defined by the Weil prime divisors of some quasi-projective normal model X of K. This result itself relies on de Jong's theory of alterations [J]. In the second part of the manuscript, we give a simplified version of a part of the "abstract non-sense" from Part II of [P4] (which follows a suggestion by Deligne [D2]), reminding one in some sense of the abstract class field theory. We define so called pre-divisorial Galois formations. The aim of this theory is to lay an axiomatic strategy for the proof of the main result. The interesting example of pre-divisorial Galois formations are the geometric Galois formations, which arise from geometry (and arithmetic). A very basic results here is Proposition 3.18, which shows that in the case K|k is a function field with td(K|k) > 1 and k an algebraic closure of a finite field, the geometric Galois formations on G = G ℓ,ab K are group theoretically encoded in G ℓ K. Finally in the last Section we prove the main Theorem announced above. The main tool here is Proposition 4.1, which gives a Galois characterization of the "rational projections". Thanks: I would like to thank several people who showed interest in this work, both for criticism and suggestions, and for careful reading. My special thanks go to