Values of differentials of plane branches: block decomposition and semiroots

Abstract

We study the passage of special gaps from a Zariski semiroot to a plane branch. A decomposition into rectangular blocks allows us to compute the maximal valuations of the coefficients of differentials and describe the image of the transition map. We treat vertical and horizontal decompositions, as well as decompositions with alternations, obtaining criteria for the preservation of special gaps. For towers whose ramification indices after the first are all equal to two, the value semigroup of the branch and the special gaps of the first semiroot determine the special gaps of the branch.