Patching and Thickening Problems
Patching and Thickening Problems
复制标题
修补和加厚问题
DOI：
10.1006/jabr.1998.7574
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发表时间：
1999
影响因子：
0.9
通讯作者：
Katherine F. Stevenson
中科院分区：
文献类型：
--
作者：
D. Harbater;Katherine F. Stevenson
近年来，在形式几何和刚性几何中，利用补片技术研究代数曲线基本群的结构取得了很大进展。其中一些结果涉及到特征为p的代数闭域上的曲线，例如Abhyankar猜想的证明（[Ra], [Ha5]）和Shafarevich猜想的几何情形的证明（[Po1], [Ha6]），以及投影曲线上伽罗瓦群的实现（[Sa1], [St1]）。虽然严格的修补方法通常被认为比正式的方法更直观，但它的基础并不完善。但是，涉及正式方法的构造往往在技术上更麻烦。本文的目的是建立在以前的正式修补结果的基础上，以便创建一个框架，在这个框架中，这样的结构是容易的。在此过程中，我们证明了一个结果，即域k上的奇异曲线可以在奇异轨迹的形式邻域内加厚为k[[t]]上具有规定行为的曲线，对于曲线的覆盖也是如此。然后，我们得到了在大域上曲线基本群的应用。本文的结构如下：第1节讨论幂级数环R = k[[t1，…]上投影曲线X *的补片问题。tn]]。证明了（定理1）在X上给出一个相干射影模等价于在闭合纤维X的每个奇点的形式邻域上以及在X的奇异轨迹S的补上的形式增厚上相容地给出这样的模。第2节将此应用于绝对和相对意义上的增稠问题。也就是说，它证明（定理3）这样的X∗可以由它的闭纤维X和S附近的完全局部增厚构造，使得X∗是远离S的平凡变形。此外（定理2），给定一个态射X→Z和Z的增厚Z∗，使得S附近的局部增厚与Z的增厚相容，则存在一个唯一的X的增厚X∗与给定的数据相容。在第3节中，这些结果应用于加厚和变形覆盖的问题。定理4结合了第1节和第2节的结果，表明可约k曲线的覆盖可以加厚到k[[t]]以上的曲线的覆盖，定理5然后展示了如何在大域上使用它来
In recent years, much progress has been made on the structure of fundamental groups of algebraic curves by means of patching techniques in formal and rigid geometry. A number of these results have concerned curves over algebraically closed fields of characteristic p — e.g. the proofs of the Abhyankar Conjecture ([Ra], [Ha5]) and of the geometric case of the Shafarevich Conjecture ([Po1], [Ha6]), and the realization of Galois groups over projective curves ([Sa1], [St1]). While the rigid approach to patching is often regarded as more intuitive than the formal approach, its foundations are less well established. But constructions involving the formal approach have tended to be technically more cumbersome. The purpose of the current paper is to build on previous formal patching results in order to create a framework in which such constructions are facilitated. In the process we prove a result asserting that singular curves over a field k can be thickened to curves over k[[t]] with prescribed behavior in a formal neighborhood of the singular locus, and similarly for covers of curves. Afterwards, we obtain applications to fundamental groups of curves over large fields. The structure of the paper is as follows: Section 1 concerns patching problems for projective curves X∗ over a power series ring R = k[[t1, . . . , tn]]. It is shown (Theorem 1) that giving a coherent projective module over X is equivalent to giving such modules compatibly on a formal neighborhood of each singular point of the closed fibre X , and on the formal thickening along the complement of the singular locus S of X . Section 2 applies this to thickening problems, in both the absolute and relative senses. Namely, it shows (Theorem 3) that such an X∗ can be constructed from its closed fibre X and from complete local thickenings near S, such that X∗ is a trivial deformation away from S. Moreover (Theorem 2), given a morphism X → Z and a thickening Z∗ of Z, such that the local thickenings near S are compatible with that of Z, there is a unique such thickening X∗ of X that is compatible with the given data. In Section 3 these results are applied to the problem of thickening and deforming covers. Theorem 4 there combines the results of Sections 1 and 2 to show that covers of reducible k-curves can be thickened to covers of curves over k[[t]], and Theorem 5 then shows how this can be used over large fields to