Geometry of singularities of mapping II

我们考虑喷流空间中的Thom-Bordman流形$Sigma^{i，j}$(及其Zariski闭包)。我们讨论了Cohen-Macaulay性质(从代数几何的交集理论的观点来判断它是否是Cohen-Macaulay性质是一个大问题)。它是由Ronga先生的去单一化和支持$Sigma^{n^-p+1，1}$的构造复杂性组成的。给出了当n=2，3，4时，映射芽$(C^n，0)到(C^2，0)$稳定扰动中出现的尖点个数的显式计算公式。我们还讨论了在此背景下出现的几个微分方程。上一段中的Thom-Boardman子流形的概念对分析这一点起到了关键的作用。舍入和展平的概念是这样定义的。我们还讨论了如果这些点是孤立的，如何定义指数。我们注意到，我们可以捆绑完全相同的…如果不是，则子流形有奇点的可能性更大。我们还得到了对于秩为1的映射$g：R^2到R^3$(对于这样的映射数，舍入指数是1或更小)的Lowner猜想的一个类比。此外，还将几个二元微分方程化为主线微分方程的推广，定义了全实的概念，研究了指数的基本性质，讨论了它们的奇点分类。(下一页t的第三篇论文)考虑函数对函数水平的限制是一个有趣的问题。当后一类函数的水平可并行化时，可以自然地定义某一映射，证明其映射度是正点轨迹和负点轨迹的欧拉特征标志的差值。讨论了可并行化的充要条件。这与引数结构和Carey结构有关。此外，我们还给出了关于圆弧分析图(下一页第二篇)的逆映射定理和吹风分析图理论(下一页第六篇)的最新进展。较少

We consider Thom-Bordman manifolds $Sigma^{i, j}$ (and their Zariski closure) in the jet space. We discussed the Cohen-Macaulay property (It is a big problem to decide whether it is Cohen-Macaulay or not from the view point in intersection theory in algebraic geometry.). It is composed by Mr.Ronga's desingularization and construct complecies supported $Sigma^{n^-p+1,1}$. An explicit formula counting the number of cusps which appeared in stable perturbation of the map-germ $(C^n,0) to(C^2,0)$ when n=2,3,4. (The first paper in the next page)Next, a classical differential geometry is discussed from the singular view point, significant point theory. We also discuss several differential equation appeared in this context. The notion of Thom-Boardman submanifold in the foregoing paragraph plays key rule to analyse this.. The notion of rounding and flattening are defined in this way. We also discuss to define the index if these points are isolated. We remark that we can bundle exactly the same … More way if do not the submanifold has singularities.We also obtain an analogy of Lowner's conjecture for rank 1 map $g : R^2to R^3$ (The rounding index is 1 or less for such maps number). Moreover, several differential equation of two variables was caught as generalization of the differential equation of principal line, and define the notion of totally real and investigate the fundamental properties of index, classification of their singularities are discussed. (the third paper in the next page t)It is interesting problem to consider the restriction of function to the level of function. A certain map can be naturally defined when the levels of the later function are parallelizable It is shown that its mapping degree are differences of Euler characteristics signposts of a positive point locus and negative point locus. The necessary and sufficient condition for parallelizability was also discussed. This is related with quotanion number structure and the Carey structures. (the fourth paper in next page)Additionally, we show the inverse-map theorem concerning the arc analysis map (the second paper in the next page) and the recent progress of the theory of the blow analysis map (the sixth paper in the next page). Less