论文标题
奇异曲线的GALOIS覆盖率
Galois Covers of Singular Curves in Positive Characteristics
论文作者
论文摘要
我们研究了在任意主要特征的代数封闭场上定义的奇异降低连接曲线的典型基本组。结果表明，当曲线是投影曲线时，étale基本组是其正常化的典型基本组的免费产物，其自由有限生成的profinite组的等级已确定。由于该结果和平滑案例的已知结果，有限群体出现作为典型基本组的商的必要条件。接下来，我们为Aggine积分曲线$ U $提供类似的结果。我们提供了一个完整的理论分类，其中有限群体作为Galoisétale连接的$ U $的Galois组。 In fact, when $U$ is a seminormal curve embedded in a connected seminormal curve $X$ such that $X - U$ consists of smooth points, the tame fundamental group $π_1^t(U \subset X)$ is shown to be isomorphic to a free product of the tame fundamental group of the normalization of $U$ with a free finitely generated profinite group whose rank is known.对于某些奇异曲线，也提出了惯性猜想的类似物。
We study the étale fundamental groups of singular reduced connected curves defined over an algebraically closed field of arbitrary prime characteristic. It is shown that when the curve is projective, the étale fundamental group is a free product of the étale fundamental group of its normalization with a free finitely generated profinite group whose rank is well determined. As a consequence of this result and the known results for the smooth case, necessary conditions are given for a finite group to appear as a quotient of the étale fundamental group. Next, we provide similar results for an affine integral curve $U$. We provide a complete group theoretic classification on which finite groups occur as the Galois groups for Galois étale connected covers of $U$. In fact, when $U$ is a seminormal curve embedded in a connected seminormal curve $X$ such that $X - U$ consists of smooth points, the tame fundamental group $π_1^t(U \subset X)$ is shown to be isomorphic to a free product of the tame fundamental group of the normalization of $U$ with a free finitely generated profinite group whose rank is known. An analogue of the Inertia Conjecture is also posed for certain singular curves.
