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

Serre, Jean-Pierre

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].

Reviewer: A.Gimigliano (Firenze)

MSC:

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

Keywords:

structure of fundamental group; Galois coverings of a given curve

Citations:

Zbl 0798.14013