On the fundamental group of a unirational 3-fold
On the fundamental group of a unirational 3-fold
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关于无理三重的基本群
DOI：
10.1007/bf01389903
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发表时间：
1978
影响因子：
3.1
通讯作者：
N. Nygaard
中科院分区：
文献类型：
--
作者：
N. Nygaard
本文的目的是证明：如果X是定义在代数闭域k上的单有理3重群，则代数基本群nj（X）为零。维数3是第一个非平凡的情况，因为单有理曲线和曲面是有理的（根据Ltiroth定理和Castelnuovo准则），因此是单连通的。Serre在他的论文[12]中指出，如果X ~ x，Y是n次的6层覆盖，则Riemann-Roch定理意味着X（Cx）= nX（Or），所以如果X（~ x）= 1，则覆盖是平凡的。单有理簇的基本群是有限的（[4]），单有理簇的任何6 tale覆盖也是单有理的，这一事实暗示了存在一个单有理的极大6 tale覆盖，因此问题简化为证明X（Cx）= 1，其中X是单连通单有理簇。
The aim of this paper is to show that if X is a unirational 3-fold defined over an algebraically closed field k, then the algebraic fundamental group nj (X) is zero. Dimension 3 is the first non-trivial case since unirational curves and surfaces are rational (by Ltiroth's theorem and Castelnuovo's criterion) and consequently simply-connected. On the other hand there are unirational 3-folds that are not rational ([1]).In his paper [12] Serre observed that if X--, Y is an 6tale covering of degree n then the Riemann-Roch theorem implies that X (Cx)= nX (Or), so if X (~ x)= 1 the covering is trivial. The fact that the fundamental group of a unirational variety is finite ([4]) and the fact that any 6tale cover of a unirational variety is again unirational implies the existence of a maximal 6tale cover which is unirational so the problem reduces to showing that X (Cx)= 1 for X a simplyconnected unirational variety.