Singularities at infinity and their vanishing cycles
Let \(f:\mathbb{C}^n \to\mathbb{C}\) be a polynomial of degree \(d\). One replaces \(f\) by a proper mapping \(t:X\to \mathbb{C}\) (which depends on the chosen system of coordinates on \(\mathbb{C}^n)\), where \(X\) is the closure in \(\mathbb{P}^n \times\mathbb{C}\) of the graph of \(f\). Let \(H_\infty\) denote the hyperplane at infinity and let \(X_\infty: =X\cap (H_\infty \times\mathbb{C})\). The least fine Whitney stratification \({\mathcal W}\) of \(X\) that contains the stratum \(X\backslash X_\infty\) is called the canonical Whitney stratification at infinity of \(X\). The authors say that the polynomial \(f\) has isolated \({\mathcal W}\)-singularities at infinity if \(t\) has isolated singularities with respect to the stratification \({\mathcal W}\). The main result of the paper is the following: Let \(f:\mathbb{C}^n \to \mathbb{C}\) be a polynomial with isolated \({\mathcal W}\)-singularities at infinity. Then the general fibre of \(f\) is homotopy equivalent to a bouquet of spheres of real dimension \(n-1\). The existence of an isolated \({\mathcal W}\)-singularity at infinity is related to the existence of a certain polar curve as germ at that point. One proves that, if the germ of such polar curve is nonvoid, then the point is a singularity where a certain positive number of cycles of a general fibre vanish.
scientific article; zbMATH DE number 871844
scientific article