Algebraization of analytic curve germs passing through isolated surface singularities
It is proved the following globalisation result for curve germs in a complex analytic isolated surface singularity. Given any isolated surface singularity \((X,0)\) and curve germs \(C_1, \dots, C_k \subset X\) it is possible to find a projective surface \(Y\) with a unique singular point \(y\) together with \(k\) curves \(D_1, \dots, D_k \subset Y\) and an isomorphism \(\varphi : (X,0) \to (Y,y)\) such that \(\varphi (C_i) = D_i\). If \((X,0)\) is a rational singularity the above result can be improved, in fact in this case we can choose \(Y\) to be independent to the curve \(C_i\). Given a pair \((X,C)\), \(X\) rational surface singularity, \(C \subset X\) reduced curve it is possible to keep track of the resolution process by a two-colored graph called the Singularity Tree, defined by extending in the natural way the classical notions of singularity tree of a germ of plane curve singularity and Dynkin diagram of a surface singularity. The singularity tree is invariant under analytic germ isomorphism and it is proved that two pairs \((X,C)\), \((X,D)\) are equivalent if they have the same singularity tree and sufficiently big intersection multiplicities \(D \cdot C_i\), with \(C_i\) irreducible components of \(C\).
scientific article; zbMATH DE number 938556
scientific article