Local geometry of planar analytic morphisms (Q2477150)

The paper is devoted to study analytic morphisms \(f: S \rightarrow T\) of smooth complex analytic surfaces locally at points \(O\) of \(S\) and \(f(O)\). Specifically, the author is interested in describing how either \(f\) or its inverse image act on singularities of curve. The main tools used in the paper to reach the above purpose are the jacobian and the main trunk of \(f\) and the tangent map to \(f\). The last two objects are introduced in the paper and a rational algorithm to compute them is given. In addition to the main trunk, further trunks are also attached to \(f\), they determine the equisingularity types of the inverse maps and Jacobian curve attached to \(f\). The main trunk of \(f\) is defined by the author as certain unibranch weighted cluster. It is proved that it determines the multiplicities of the inverse images and the ones of the Jacobian curve, it allows to bound the ratio between the multiplicities of a germ of curve and its inverse image and it gives information on the singularities of direct images. The tangent map to \(f\) is a rational map from the first neighbourhoods of \(O\) to the ones of the last point in the main trunk. In the paper, it is proved that the tangent map helps to describe the cone tangent of inverse images and that it is related with the tangent cone of the Jacobian. Finally, we add that the existence and distribution of irreducible germs of curve whose representatives are not injectively mapped onto their direct images are also treated in the paper.