Carrousel in family and non-isolated hypersurface singularities in \(\mathbb{C}^3\) (Q334472)

For a reduced holomorphic germ \(f: ({\mathbb C}^3,0)\to ({\mathbb C},0)\) one may consider the link \(L_0=f^{-1}(0)\cap {\mathbb S}^5_\epsilon\), the boundary \(L_t=f^{-1}(t)\cap {\mathbb S}^5_\epsilon\), and the link \(\bar{L}_0\) of the normalisation of \(f^{-1}(0)\cap{\mathbb B}^6_\epsilon\). For isolated singular points these are all the same diffeomorphically, but if \(f\) has a non-isolated singularity they may differ: indeed \(L_0\) is even no longer a differentiable manifold.NEWLINENEWLINEThe authors give a full proof of an assertion they made in [Int. Math. Res. Not. 2003, No. 43, 2305--2311 (2003; Zbl 1032.32018); erratum ibid. 2004, No. 6, 309--310 (2004)] that \(L_t\) is a graph manifold (as \(\bar{L}_0\) is) whose Seifert pieces have oriented basis.NEWLINENEWLINETheir method also allows them to compare the topologies of \(\bar{L}_0\) and \(L_t\). They show that these two are not homeomorphic if the singularity of \(f\) really is non-isolated, unless perhaps \(f\) is reduced and \(L_t\) is a lens space. This is a mild restriction because germs for which \(L_t\) is a lens space form a very restricted class and their resolutions can be described in some detail.NEWLINENEWLINEThe proofs are very technical but there are two important ideas, similar in that both of them carry out a standard procedure but in a family parametrised by a circle or a punctured disc. The first one is an embedded resolution result: roughly, one can resolve simultaneously by blowups the singularity of every curve section of \((f=0)\) obtained by fixing one coordinate (e.g., by setting \(x=x_0\) for \((x,y,z)\in {\mathbb C}^*\times {\mathbb C}^2\)). The second one is what the title of the paper promises: a version of the carrousel construction of \textit{Lê Dũng Tráng} [in: C.P. Ramanujam -- A tribute. Collect. Publ. of C.P. Ramanujam and Pap. in his Mem., Tata Inst. fundam. Res., Stud. Math. 8, 157--173 (1978; Zbl 0434.32010)] that works simultaneously in a family parametrised by \({\mathbb S}^1\).