Construction of covers in positive characteristic via degeneration
Proceedings of the American Mathematical Society
In this paper we construct examples of covers of the projective line in positive characteristic such that every degeneration is inseparable. The result illustrates that it is not possible to construct all covers of the generic -pointed curve of genus zero inductively from covers with a smaller number of branch points. Let be an algebraically closed field of characteristic > 0. Let = ℙ 1 and be a finite group. We fix ≥ 3 distinct points x = ( 1 , 2 , . . . , ) on . We ask whether there exists a
... her there exists a tame Galois cover : → with Galois group which is branched at the . If does not divide the order of , then the answer is well known. Namely, such a cover exists if and only if may be generated by − 1 elements. Suppose that divides the order of . Then the existence of a -cover as above depends on the position of the branch points (see, for example, [8, Lemma 6]). In this paper we restrict to the case that ( ; x) is the generic -pointed curve of genus zero. A more precise version of the existence question in positive characteristic is whether there exists a -Galois cover of ( ; x) with given ramification type (see, for example, ). For the particular kinds of groups we consider here, we define the ramification type in §1. Osserman () proves (non)existence of covers in positive characteristic, for certain ramification types. His method is roughly as follows. First, he proves results for covers branched at = 3 points. In this case his results are strongest. Using the case = 3, he then constructs admissible covers of degenerate curves which deform to covers of smooth curves (see §2 for a definition). Suppose we are given a tame -Galois cover of ( = ℙ 1 ; x). Osserman ([5, §6]) asks whether there exists a degeneration (¯,x) of ( ; x) such that specializes to an admissible cover of (¯,x). If such a degeneration exists, he says that has a good degeneration. Covers which admit a good degeneration are exactly those which may be shown to exist inductively from the existence of covers with fewer branch points. The goal of this paper is to produce covers which do not have a good degeneration. We show that such covers exist with an arbitrarily large number of branch points.