## Coverings of algebraic curves. (Revêtements de courbes algébriques.)(French)Zbl 0798.14014

Séminaire Bourbaki, Vol. 1991/92. Exposés 745-759 (avec table par noms d’auteurs de 1948/49 à 1991/92). Paris: Société Mathématique de France, Astérisque. 206, 167-182 (Exp. No. 749) (1992).
The paper is a survey of the results about the problem of determining the Galois coverings of a given curve: More precisely, let $$k$$ be an algebraically closed field, $$C$$ an integral, smooth algebraic curve on $$k$$ (so $$C= \overline{C} -S$$, where $$\overline C$$ is a smooth projective curve and $$S$$ a finite subset of $$\overline C$$:
(a) For a given finite group $$G$$, when does there exist a connected Galois covering $$C' \to C$$ whose Galois group is $$G$$?
(b) Let $$\pi_ C = \pi_ 1^{\text{alg}} (C,x)$$ the (algebraic) fundamental group of $$C$$ with respect to a base-point $$x$$. What is the structure of $$\pi_ C$$?
Note that (b) is a stronger question since a profinite group $$G$$ satisfies (a) if and only if it is a quotient of $$\pi_ C$$.
If $$k = \mathbb{C}$$ then the answers are known and they are also valid if $$\text{char} k = 0$$. A. Weil suggested that the theory should be the same in case $$\text{char} k = p > 0$$ if $$p$$ does not divide the order of the group; this idea was made in a more precise conjecture by Abhyankar in 1956, and proved by Groethendieck in 1958. When $$C$$ is complete (i.e. $$S = \emptyset)$$, the answer is known for $$g(C) = 0,1$$; while there are not even conjectures if $$g \geq 2$$ (there are results about the cohomology of $$\pi_ C)$$.
In the affine case, let $$| S | = s$$ and $$p(G)$$ be the subgroup generated by the Sylow $$p$$-subgroups of $$G$$, then Abhyankar conjectured: A finite group $$G$$ is a quotient of $$\pi_ C$$ if and only if $$G/p (G)$$ can be generated by $$2g + s - 1$$ elements.
In the case when $$C$$ is a line $$g=0$$ and $$s=1$$, so $$2g + s - 1 = 0$$ and this conjecture says that $$G$$ is a quotient of $$\pi_ C$$ if and only if $$G$$ is generated by its Sylow $$p$$-subgroups. Examples and partial results about this case are given; the conjecture has been later proved by M. Raynaud [Invent. Math. 116, No. 1-3, 425-462 (1994; see the preceding review)].
For the entire collection see [Zbl 0772.00016].

### MSC:

 14H30 Coverings of curves, fundamental group 14F35 Homotopy theory and fundamental groups in algebraic geometry

Zbl 0798.14013
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