The Nash modification and hyperplane sections on surfaces (Q2583495)

The author proves that the normalized Nash modification [see for instance \textit{M. Spivakovsky}, Ann. Math. (2) 131, No. 3, 411--491 (1990; Zbl 0719.14005)], of a reduced equidimensional surface singularity factors through the blow-up if and only if its reduced tangent cone does not contain any two dimensional plane. For the proof the author uses the normal-conormal diagram adapted to the Nash modification [see \textit{Lê Dung Tráng} and \textit{B. Teissier}, Comment. Math. Helv. 63, No. 4, 540--578 (1988; Zbl 0658.32010)]. Then he uses his previous results in [\textit{J. Snoussi}, Comment. Math. Helv. 76, No. 1, 61--88 (2001; Zbl 0990.32005)] and proves that for a normal singularity the normalized blow-up of the origin dominates the normalized Nash modification if and only if they are isomorphic. In this case they both desingularize the singularity. Some examples at the end are presented.