The singularities of algebraic and analytic varieties constitute an old and today very active field of research which combines techniques and viewpoints from different mathematical fields such as Geometry, Algebra, Topology and Function Theory. Doubtless, the oldest and best understood singularities are those of plane curves. Some curves with singular points appear in the work of the ancient Greek geometers, and the first contribution to a systematic study of plane curve singularities is due to Isaac Newton. Even if some relevant questions still remain open, nowadays, after the work of geometers such as Puiseux, Smith, Noether, Halphen, Enriques and Zariski, there is a well established theory for the analysis and classification of the singularities of plane curves. I have intended in this book to give a precise and detailed account of most of the main facts of this theory from a decidedly geometrical viewpoint which, I hope, will not completely hide its more algebraic or topological aspects. Infinitely near points have been taken as the cornerstone of the presentation: main description and classification of singularities are stated using infinitely near points, and other properties and invariants, such as characteristic exponents, Milnor number, discriminantal index, semigroups, polar quotients, etc., are related to them.

Infinitely near points are a nice and old idea for describing singularities. They appear in the work of M. Noether and their geometry was extensively developed by Enriques ([35], Book IV). Since then the geometry of infinitely near points has seldom appeared in the mathematical literature, maybe due to the rather obscure and non-intrinsic way the Italians used to introduce infinitely near points, or also to the fact that they do not have a straightforward translation in terms of rings and ideals.