On some conjuctures in singularity theory ; Sobre algunas conjeturas en teoría de singularidades

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Álgebra, Geometría y Topología, leída el 21-11-2019 ; This thesis is dedicated to the study of singular points of flat curves. Specifically, we provide counterexamples to some of the conjectures that have arisen following the investigation of the singularities of these curves. We deal with two main themes: The conjecture of the monodromy of J. Denef and F. Loeser and its generalization by A. Némethi and W. Veys. Free and almost-free flat curves with isolated singularities, and some related conjectures proposed by A Dimca and G. Sticlaru. The monodromy conjecture was demonstrated for flat curves by F. Loeser in [Loe88], and later extended for zeta functions associated with arbitrary differential forms by W. Veys. Subsequently A. Némethi and W. Veys introduced the set of allowed differential forms, which contains the standard differential form. In this context, it is worth asking whether there could be another differential form defined in a naturally allowed way. One choice could be the Hessian differential form, since the poles of the zeta function associated with this differential form give rise to self-values of the monodromy in numerous examples. However, Chapter 3 demonstrates that Hessian is not a permitted differential form . ; Esta tesis está dedicada al estudio de puntos singulares de curvas planas. En concreto, proporcionamos contraejemplos a algunas de las conjeturas que han surgido a raíz de la investigación de las singularidades de dichas curvas. Tratamos dos temas principales:¿La conjetura de la monodromía de J. Denef y F. Loeser y su generalización por A. Némethi y W. Veys.¿Curvas planas libres y casi-libres con singularidades aisladas, y algunas conjeturas relacionadas propuestas por A. Dimca y G. Sticlaru.La conjetura de la monodromía fue demostrada para curvas planas por F. Loeser en [Loe88], y extendida más tarde para funciones zeta asociadas a formas diferenciales arbitrarias por W. Veys. ...