Includes bibliographical references (p. 357-367) and index.

Preface -- 1. Preliminaries -- 2. Puiseux' theorem -- 3. Resolutions -- 4. Contact of two branches -- 5. Topology of the singularity link -- 6. The. Milnor fibration -- 7. Projective curves and their duals -- 8. Combinatorics on a resolution tree -- 9. Decomposition of the link complement and the Milnor fibre -- 10. The. monodromy and the Seifert form -- 11. Ideals and clusters -- References -- Index.

The simplest singularities of a plane curve are self-crossings and cusps. Equivalence of singular points of (complex) plane curves can be defined using combinatorial data, resolution data or topological data; all give the same result. The first half of this book, which is based on a M.Sc. Course, works up to this synthesis via Puiseux series (parametrising the curve), resolution of singularities, infintely near points and the Alexander polynomial. For curves in the projective plane, formulae for the genus and the class depend on the singularities. The topology gives a fibration (due to Minor), described by the monodromy self-map of the fibre, a compact surface. The monodromy is approached through resolution trees, the group of exceptional cycles, combinatorial data, and the decomposition theorems of Thurston and of Jaco-Shalen-Johannsen. The author obtains a criterion for the monodromy to be (setwise) finite and a close relation between the Eggers tree, the resolution graph and the Eisenbud-Neumann diagram. Ahe calculates the characteristic polynomials of the monodromy, studies the Seifert form, and calculates the signatures that determine it over the reals. Ideals in the local ring of appoint are related to the cycles studied earlier, and (by a Galois correspondence) with the clusters of Enriques; this involves valuative and integral closure. This graduate text gives an introduction to this attractive area of mathematics. By synthesizing different perspectives it offers a novel view and a number of new results. Exercises and suggestions for further rreading are included. - Back cover.

All Ages.