Combinatorics and their evolution in resolution of embedded algebroid surfaces (Q2657839)

This paper studies the embedded resolution of an algebroid surface over an algebraically closed field of characteristic zero, that is the spectrum of a ring \( K[[X, Y,Z]]/(F)\). The main combinatorial object associated to \(F\) is Hironaka's characteristic polygon \(\Delta(F)\). The original motivation of this work is: can the combinatorics bound, in some effective sense, the resolution process? The paper studies in detail the resolution process for prepared equations, that \(F\) is a generic Weierstrass-Tchirnhausen equation, of the form \(Z^n +\sum_{k=0}^{n-2}a_k(X,Y)Z^k\) with \(a_k\) regular in \(X\) of order \(\nu_k=\nu(a_k)\geq n-k\). The resolution strategy used is the following: (1) if \((Z,X)\) or \((Z, Y )\) are permissible curves, a monoidal transformation centered at them is performed, (2) otherwise, a quadratic transformation. For prepared equations bounds are given for the number of blow-ups needed before the multiplicity drops. The paper contains many examples.