On the Mordell-Weil lattice of the elliptic curve y^2=x^3+t^m+1(IV)
On the Mordell-Weil lattice of the elliptic curve y^2=x^3+t^m+1(IV)
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在椭圆曲线的 Mordell-Weil 格子上 y^2=x^3 t^m 1(IV)
DOI：
10.1007/bfb0086194
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发表时间：
2008
影响因子：
5.6
通讯作者：
Hisashi Usui
中科院分区：
文献类型：
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作者：
Hisashi Usui
在我们在京都研讨会上的发言中，我们引入了椭圆曲面的Mordell-Weil格的概念，并解释了有理椭圆曲面的Mordell-Weil格的基本结果(见[S1])，在这篇笔记中，我们讨论了这一理论在奇点形变研究中的应用。从这个新的观点出发，我们可以更全面、更精细地反驳Briekorn、Tjuina等人(参看[B1，2]、[Dpt]、[L]、[O]、[Sl])关于ES‘E7或Es’型有理重点形变的已知结果。特别地，我们给出了单调满射性的纯代数证明，即参数空间中判别轨迹的补的基本群映射到Weyl群W(Er)上。此外，我们根据奇点的类型得到了参数空间分层的非常精确的描述。
O. Introduction In our talk at the Kyoto Symposium, August 19S9, we introduced the notion of the Mordell-Weil lattice of an elliptic surface and explained the basic results on the Mordell-Weil lattices of rational elliptic surfaces (see [S1] for the summary), In this note, we discuss an application of this theory to the study of deformation of singularities. From this new viewpoint, we can reprove the well-known results on the deformation of rational double points of type ES'E7or Es' due to Briekorn, Tjurina and others (cf.[B1, 2],[DPT],[L],[O],[Sl]), in a more global, refined form. In particular, we have a purely algebraic proof of the surjectivity of the monodromy, which says that the fundamental group of the complement of the discriminant locus in the parameter space maps onto the Weyl group W (E r). Further we obtain a very precise description of the stratification of the parameter space according to the type of singularities.