Let \(S, S'\) be two germs of smooth analytic complex surfaces at distinguished points \(o, o',\) respectively, let \(\varphi: S'\rightarrow S\) be an analytic morphism, and let \(\Delta\) be the discriminant of \(\varphi.\) The author proves that the Milnor numbers of \(\xi,\) the germ of a curve in \(S,\) and the inverse image \(\varphi^\ast(\xi)\) are related as follows: \(\mu(\varphi^\ast(\xi))= \kappa (\mu(\xi)-1) + [\xi.\Delta] + 1,\) where \(\kappa\) is the local degree of the map \(\varphi\) at \(o',\) and the square brackets denote the intersection multiplicity. The proof is based essentially on the Milnor formula \(\mu = 2\delta - r + 1\) [J. Milnor, Singular points of complex hypersurfaces, Ann. Math. Stud. 61 (1968; Zbl 0184.48405)].
Then the author shows that the inverse images of reduced germs \(\xi_1\) and \(\xi_2\) are equisingular when they are in the same position relative to the discriminant \(\Delta\) and have no common branches with it. The first condition is expressed in terms of the intersection multiplicities of branches \(\xi_1,\) \(\xi_2\) and \(\Delta.\)