Covers of the Affine Line in Positive Characteristic with Prescribed Ramification
Covers of the Affine Line in Positive Characteristic with Prescribed Ramification
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具有规定分枝的正特征仿射线的覆盖
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发表时间：
2011
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通讯作者：
I. Bouw
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作者：
I. Bouw
令 k 为特征 p > 0 的代数闭域。在本文中，我们考虑伽罗瓦覆盖 g : Y → P1k ，其仅在 t = ∞ 处分支。我们将这种覆盖称为仿射线的未分支覆盖。根据 Raynaud ([7]) 证明的 Abhyankar 对仿射线的猜想，当且仅当 G 是拟 p 群，即可以由 p 幂阶的元素生成时，对于给定群 G 存在这样的覆盖。许多群满足这个性质，例如所有可被 p 整除的单阶群都是拟 p 群。对于给定的拟 p 群 G，仿射线有许多未分支的覆盖。例如，如果 G = Z/pZ，则此类覆盖层有无限多个系列。在这篇文章中，我们修复了一个拟 p 群 G，并询问仿射线的未分支 G-伽罗瓦覆盖的最小属是什么。这个问题是由 Muskat-Pries 的工作引发的（[6]）。 Muskat 和 Pries 计算了 Abhyankar 发现的许多交替群覆盖的属 ([1])。事实证明，他们认为的封面在某种意义上具有“小”属，可以使其精确。我们证明，对于 p + 2 ≤ d < 2p，[6] 的覆盖确实是最小属的仿射线的未分支的 Ad 覆盖。研究具有给定分支的覆盖的存在性的另一个动机来自稳定约简理论。证明正特征中存在具有给定驯服分支的覆盖的一种方法是证明并非所有覆盖都对特征 p 具有不好的约简。与还原不良的盖相关联的是一组尾盖。本质上，这些是对覆盖稳定还原的某些不可还原分量的限制。 Wewers ([9]) 提出的一种技术有时允许人们根据对这些尾部覆盖的充分了解来计算不良减少的覆盖数量。该技术已在 [4] 中应用于 p 级覆盖，这些覆盖在 4 个点处驯服地分支。为了将[4]的结果推广到p < d < 2p度，需要知道小属的d度仿射线覆盖的存在性。我们主要结果的证明策略如下。我们不考虑固定伽罗瓦群的覆盖，而是考虑非伽罗瓦仿射线的非分支覆盖 f : X → P。令 d 为 f 的次数。作为黎曼-赫尔维茨公式在正特征中的应用，我们根据 f 的分枝计算 f 的伽罗瓦闭包的属
Let k be an algebraically closed field of characteristic p > 0. In this note we consider Galois covers g : Y → P1k which are only branched at t = ∞. We call such covers unramified covers of the affine line. It follows from Abhyankar’s conjecture for the affine line proved by Raynaud ([7]) that such a cover exists for a given group G if and only if G is a quasi-p group, i.e. can be generated by elements of p-power order. Many groups satisfy this property, for example all simple groups of order divisible by p are quasi-p groups. For a given quasi-p group G there are many unramified covers of the affine line. For example, if G = Z/pZ there are infinitely many families of such covers. In this note we fix a quasi-p group G and ask what the minimal genus of an unramified G-Galois cover of the affine line is. This question has been motivated by the work of Muskat–Pries ([6]). Muskat and Pries compute the genus of many alternating-group covers which had been found by Abhyankar ([1]). It turns out that the covers they consider have “small” genus in a sense which can be made precise. We show that for p + 2 ≤ d < 2p the covers of [6] are indeed the unramified Ad-covers of the affine line of minimal genus. Another motivation for studying the existence of covers with given ramification comes from the theory of stable reduction. One method for showing that covers with given tame ramification exist in positive characteristic is to show that not all covers have bad reduction to characteristic p. Associated with a cover with bad reduction is a set of tail covers. Essentially these are the restriction to certain irreducible components of the stable reduction of the cover. A technique due to Wewers ([9]) sometimes allows one to count the number of covers with bad reduction in terms of sufficient knowledge of these tail covers. This technique has been applied in [4] to covers of degree p which are tamely branched at 4 points. To generalize the result of [4] to degree p < d < 2p one needs to know the existence of covers of the affine line of degree d of small genus. The strategy of the proof of our main result is as follows. Rather than considering covers with fixed Galois group, we consider unramified covers f : X → P of the affine line which are non-Galois. Let d be the degree of f . As an application of the Riemann–Hurwitz formula in positive characteristic, we compute the genus of the Galois closure of f in terms of the ramification of f