On isolated singularities of surfaces which do not affect the conditions of adjunction (Part II.)
On isolated singularities of surfaces which do not affect the conditions of adjunction (Part II.)
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关于不影响附加条件的表面的孤立奇点（第二部分）
DOI：
10.1017/s0305004100012706
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发表时间：
1934
影响因子：
0.8
通讯作者：
P. D. Val
中科院分区：
文献类型：
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作者：
P. D. Val
在本文的第一部分，我研究了孤立奇点的性质，这些奇点可以出现在代数曲面上，而不影响伴随条件和曲面的算术亏格。结果表明，这类点的邻域可分析为连通的曲线链或树，每条曲线都是有理的且虚拟次数为 -2，它们要么排列成单链（形成一个二重点，如果只有一条曲线则为圆锥节点），要么分别排列成三条链，比如分别有\(n\)、\(p\)、\(q\)条曲线，每条链的一端曲线与我们可称为树的中心曲线相交。\(n\)、\(p\)、\(q\)的值不是任意的，而是要么满足\(p = q = 1\)，要么满足\(n\leq4\)，\(p = 2\)，\(q = 1\)；前一种情况给出的单节点我们称为\(U\)，后一种情况给出的单节点我们称为\(U\)，并带有一个下标表示每种情况下类的减少。我指出了这些结果与考克斯特对反射生成群的枚举之间的相似性，这些奇点与考克斯特群之间存在一一对应关系，且有一个限制，即我们所关注的唯一群是那些任意两个对称素数之间的夹角为\(\frac{\pi}{2}\)或\(\frac{\pi}{3}\)的群。实际上，构成一个完整邻域的曲线对应于一个基本区域的边界素数，不相交的两条曲线对应于相互垂直的素数，相交的两条曲线对应于夹角为\(\frac{\pi}{3}\)的素数。
In the first part of this paper I investigated the nature of the isolated singular points which can appear on an algebraic surface without affecting the conditions of adjunction and the arithmetic genus of the surface. It appeared that such points have neighbourhoods analysable into connected chains or trees of curves, each rational and of virtual grade − 2, and arranged either in a single chain (giving a binode, or a conic node if there is only one curve) or in three chains of, say, n, p, q curves respectively, one end curve of each chain meeting one which we may call the central curve of the tree. The values of n, p, q are not arbitrary, but satisfy either p = q = 1, or n ≤ 4, p = 2, q = 1; the unodes given by the former case we call U, those given by the latter case U, with a suffix indicating the reduction in class in each case. I pointed out the similarity between these results and Coxeter's enumeration of groups generated by reflexions, there being a one-one correspondence between these singularities and Coxeter's groups, with the restriction that the only groups with which we are concerned are those in which any two primes of symmetry are inclined at either π/2 or π/3. In fact, the curves which make up a complete neighbourhood correspond to the bounding primes of a fundamental region, two curves which do not meet corresponding to mutually perpendicular, primes, and two which do to primes inclined at π/3.