Discriminant of a Germ Φ: (C² , 0)→(C² , 0) and Seifert Fibred Manifolds

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“…Moreover, we have from [Ma1], Theorem 1 and [Ma2], Theorem 1: THEOREM 6. The set fq f g ðaÞg where a is a rupture vertex of Gðp; fgÞ is equal to the set fLðK f ; nÞ=LðK g ; nÞg, where LðÀ; ÀÞ denotes the linking number in S 3 E and n is a leaf of a Seifert manifold of the minimal Waldhausen decomposition of the complement in S 3 E of K fg .…”

“…In fact, in [Ma2] the above Theorem is given in terms of Waldhausen manifolds. Indeed, it is well known that there exists a bijective correspondence between the rupture vertices of Gðp; fgÞ and the Seifert manifolds of the minimal Waldhausen decomposition of the complement in S 3 E (it is the sphere of radius E, small enough, centered at the origin of C 2 ) of the link…”

“…In [Ma2] (Proposition 3 and Theorem 7) we prove that for each i, (1 4 i 4 p), there exists at least one branch g of J such that g intersects W i . So, in terms of rupture zones it gives Theorem 5.…”

“…The dual graph, with a suitable weight for each vertex (for example one can take the number of blowing-ups done to produce the corresponding divisor or, alternatively, the self-intersection of the divisor), determines the topology of the singularity associated to fg. If, moreover, we distinguish the branches of ff ¼ 0g from those of fg ¼ 0g (for example by using different colors or different symbols for the corresponding arrows), it determines the topology of the pair ð f; gÞ ([Ma2]). …”

Discriminant d'un germe $(g, f) : (\mathbb{C}^2,0) \to (\mathbb{C}^2, 0)$ et quotients de contact dans la résolution de $f \cdot g$

“…Dans [14], nous avons défini les quotients jacobiens de (g, f ) et démontré que ce sont des invariants du type topologique de (g, f ) . Nous les avons calculés en fonction de la topologie de (g, f ).…”