Note on Cahit Arf's “Une interprétation Algébrique de la Suite des Ordres de Multiplicité D'une Branche Algébrique”*
Note on Cahit Arf's “Une interprétation Algébrique de la Suite des Ordres de Multiplicité D'une Branche Algébrique”*
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关于 Cahit Arf 的“Une interprétation Algébrique de la Suite des Ordres de Multiplicité Dune Branche Algébrique”的注释*
DOI：
10.1112/plms/s2-50.4.288
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发表时间：
1948
影响因子：
1.8
通讯作者：
P. Val
中科院分区：
文献类型：
--
作者：
P. Val
是所有超曲面与该分支的可能交集数;实际上超曲面的交集数显然是环元素的阶。如果环H是标准环，定理7表明这些交数都是分支的重数和。因此，这是典型分支的特征性质。并不像人们可能设想的那样，由此得出存在简单地通过分支的每一个连续点的超曲面;因为对应于普通尖点i1-i t 12-i的环g H= k+ k [t] t* 显然是正则的，而不存在简单地通过分支的前两个点以上的曲线。
are the possible intersection numbers of all hypersurfaces with the branch; in fact the intersection number of the hypersurface is clearly the order of the element of the ring. If the ring H is canonical, theorem 7 shows that these intersection numbers are all the multiplicity sums of the branch. This accordingly is the characteristic property of a canonical branch. It does not, as one might suppose, follow from this that there exist hypersurfaces passing simply through every number of consecutive points of the branch; for the rin g H= k+ k [t] t* corresponding to an ordinary cusp i1—i t 12—i is clearly canonical, whereas there are no curves passing simply through more than the first two points of the branch.