On the semi-universal deformation of a simple-elliptic hypersurface singularity: Part I: Unimodularity
On the semi-universal deformation of a simple-elliptic hypersurface singularity: Part I: Unimodularity
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DOI：
10.1016/0040-9383(77)90006-4
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发表时间：
1977
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--
通讯作者：
E. Looijenga
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作者：
E. Looijenga
在六阿诺德的孤立超曲面奇点的层次结构中，简单椭圆奇点紧挨着简单奇点，这一性质实际上刻画了它们。根据K.齐藤他们承认在二维以下的替代描述。设C是椭圆曲线，1是C上具有负Chern类c（1）的线丛.通过将1的零截面折叠为一点而得到的空间X 0已知允许一个标准代数结构。Saito证明了X 0是超曲面当且仅当c（1）E {--I，-2，-3}。这样得到的超曲面有一个孤立的奇点是简单椭圆的，相反，任何二维简单椭圆奇点都是以这种方式出现的（因此得名）。特别是，任何这样的奇点承认一个决议与椭圆曲线同构C作为例外因子。这种描述的优点是它解释了为什么它们出现在一维族中：除了c（l），它们通过C的j-不变量来区分，从而参数化了简单椭圆奇点的模空间。c（l）=-i（=-1，-2，-3）奇点的变形理论与仿射根系&+有着密切的联系。这就是为什么这类奇点被标记为Bgmi。(Note根据GN Tyurina和Kas-Schlessinger，具有孤立奇点的复空间允许半泛变形.本文研究了简单椭圆奇点的半普适变形。特别是，我们将在第二部分中给出一个描述拉Brieskorn-Grothendieck的判别轨迹这样的变形。本文证明了两个属于同一族的简单椭圆奇点具有拓扑等价的半万有变形。这样，我们将得到拓扑稳定的映射芽不是解析稳定的显式例子。值得注意的是，这些例子表明，无论是博德曼符号，也不是G.- M. Greuel向我指出，尖点的数目构成了稳定映射芽的拓扑不变量。这与F. Pham和其他人（cf.猜想（3.8）在暴露X的Asterisque 17）。更详细的讨论见下面的注释（1.2）。这篇论文的早期版本包含一个相当严重的错误。我感谢0。W.利亚施科和维·阿诺向我指出了这一点。
IN VI ARNOLD’S hierarchy of isolated hypersurface singularities the simple-elliptic singularities come next to the simple singularities and this property actually characterizes them. According to K. Saito they admit in dimension two the following alternative description. Let C be an elliptic curve and let 1 be a line bundle over C with negative Chern class c (l). The space X0 obtained by collapsing the null-section of 1 to a point is known to admit a canonical algebraic structure. Saito has shown that X0 is a hypersurface if and only if c (l) E {--I,-2,-3). The hypersurfaces thus obtained have an isolated singularity which is simple-elliptic and conversely any two-dimensional simple-elliptic singularity arises in this way (hence their name). In particular, any such singularity admits a resolution with an elliptic curve isomorphic to C as exceptional divisor. The advantage of this description is that it explains why they occur in one-dimensional families: apart from c (l), they are distinguished by the j-invariant of C, which thus parametrizes a moduli space of simple-elliptic singularities. The deformation theory of the singularities with c (l)=-i (=-1,-2,-3) turns out to have an intimate connection with the affine root system &+. That is why this class of singularities is labeled by the symbol Bgmi.(Note that i is just the connection index of the root system Esei).According to GN Tyurina and Kas-Schlessinger a complex space with isolated singularity admits a semi-universal deformation. We propose to investigate the semi-universal deformation of a simple-elliptic singularity. In particular, we shall give in part II a description a la Brieskorn-Grothendieck of the discriminant locus of such a deformation. The present paper will show that two simple-elliptic singularities belonging to the same family have topologically equivalent semi-universal deformations. In this way we will obtain explicit examples of topologically stable map-germs which are not analytically stable. It is noteworthy that these examples show that neither the Boardman symbol nor, as G.-M. Greuel pointed out to me, the number of cusps constitute a topological invariant for stable map-germs. This contradicts a conjecture of F. Pham and others (cf. conjecture (3.8) in expose X of Asterisque 17). For a more detailed discussion, see the remarks following (1.2). An earlier version of this paper contained a rather serious error. I thank 0. W. Liaschko and VI Arnol’d for pointing this out to me.