On the Fundamental Group of the Complement of a Node Curve
On the Fundamental Group of the Complement of a Node Curve
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论节点曲线补集的基本群
DOI：
10.2307/1971204
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发表时间：
1980
影响因子：
4.9
通讯作者：
W. Fulton
中科院分区：
文献类型：
--
作者：
W. Fulton
其中dl， * dt是C的不可约分量的度，如果知道r（p2c）中有限指数的所有子群的交集是平凡的，就会成立。Zariski最初的目标是确定P2 C的所有有限覆盖，这是根据这里所陈述的定理得出的：所显示群的无限补全是相同的，对于平缓的基本群和特征p中的素数到p补全也是如此（参见[11，[31]）。对于一般平面截面为节点曲线的PI超曲面，该定理也给出了相同的结论。Zariski的证明依赖于Enriques和Severi的断言，即任何节点曲线都可以退化为一般位置上的直线。然而，这种说法尚未得到证实。Abhyankar已经证明了这个结果的许多特殊情况。最近与J. Hansen[21]联合证明的Bertini连通性定理的强化，产生了一个引理，它完成了Abhyankar的美丽论证。设f: X -+ Y是代数闭域上射影曲面的有限态射，Y非奇异，X法向。假设f是a
where dl, * dt are the degrees of the irreducible components of C, would follow if one knew that the intersection of all subgroups of finite index in r(P 2C) were trivial. Zariski's original goal of determining all finite coverings of P2 C follows from the theorem stated here: the profinite completions of the displayed groups are the same, and similarly for the tame fundamental group and the prime-to-p completion in characteristic p (cf. [11, [31). The theorem also implies the same conclusion for a hypersurface in PI whose generic plane sections are node curves. Zariski's proof relied on the assertion of Enriques and Severi that any node curve can be degenerated to lines in general position. This assertion remains unproved, however. Abhyankar has proved many special cases of this result. A recent strengthening of Bertini's connectedness theorem, proved jointly with J. Hansen [21, yields a lemma which completes Abhyankar's beautiful argument. Let f: X -+ Y be a finite morphism of projective surfaces over an algebraically closed field, with Y non-singular and X normal. Assume f is a