Download e-book for iPad: Milnor fiber boundary of a non-isolated surface singularity by András Némethi

Algebraic Geometry

By András Némethi

In the learn of algebraic/analytic types a key element is the outline of the invariants in their singularities. This e-book ambitions the tough non-isolated case. allow f be a fancy analytic hypersurface germ in 3 variables whose 0 set has a 1-dimensional singular locus. We increase an specific technique and set of rules that describe the boundary M of the Milnor fiber of f as an orientated plumbed 3-manifold. this system additionally offers the attribute polynomial of the algebraic monodromy. We then make certain the multiplicity process of the open publication decomposition of M lower out via the argument of g for any complicated analytic germ g such that the pair (f,g) is an ICIS. additionally, the horizontal and vertical monodromies of the transversal sort singularities linked to the singular locus of f and of the ICIS (f,g) also are defined. the idea is supported through an excessive amount of examples, together with homogeneous and composed singularities and suspensions. The homes odd to M also are emphasized.

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By blowing up the leftedge, and blowing down the strict transform of the 0-vertex, we realize that e can be 34 4 Plumbing Graphs and Oriented Plumbed 3-Manifolds replaced by e ˙ 1. Hence by repeating this pair of operations, we can reduce e to 0. Then a 0-chain absorption of this newly created 0-vertex finishes the argument. In this book we will manipulate only the operations listed above. Nevertheless, if the reader wishes to use some other operations of the (oriented) plumbing calculus, this is perfectly fine if she or he wishes to focus only on @F .

0/ D ff D 0g \ fg D 0g has only an isolated singularity at the origin. Vf // \ Vg D f0g and Vf intersects Vg in the complement of the origin transversally in a smooth punctured curve. 2. C3 ; 0/ ! C; 0/ be an analytic germ with 1-dimensional critical locus. C3 ; 0/ ! f; g/ forms an ICIS. f; g/. C3 ; 0/, where ˚ is not a local submersion. Its image . C2 ; 0/ is called the discriminant locus of ˚. 3. B] Fix a germ ˚ as above. Then there exist a sufficiently small closed ball B C3 of radius , and a bidisc DÁ2 C2 with radius 0 < Á such that: 1.

KX ; Vf /. C; 0/; which is not necessarily an isolated singularity. Set Kf WDKX \Vf . X; f / and the generalized Milnor fibration arg WD f =jf j W KX n Kf ! S 1 . If f defines an isolated singularity, then this is an open book of KX with binding Kf . 4. The fibration arg W KX n Kf ! KX nKf ; Z/ ! Z. arg /, then the page of arg has n connected components which are cyclically permuted by the monodromy. The map arg can be compared with the multiplicity system as follows. Let ZV be the free abelian group generated by fhvigv2V .

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