Singular Points of Plane Curves (London Mathematical Society Student Texts, Series Number 63)

Even the simplest singularities of planar curves, e.g. where the curve crosses itself, or where it forms a cusp, are best understood in terms of complex numbers. The full treatment uses techniques from algebra, algebraic geometry, complex analysis and topology and makes an attractive chapter of mathematics, which can be used as an introduction to any of these topics, or to singularity theory in higher dimensions. This book is designed as an introduction for graduate students and draws on the author's experience of teaching MSc courses; moreover, by synthesising different perspectives, he gives a novel view of the subject, and a number of new results. Contents......Page 7 Preface......Page 11 1.1 What is a plane curve?......Page 15 1.2 Intersection numbers......Page 20 1.3 Resultants and discriminants......Page 22 1.4 Manifolds and the Implicit Function Theorem......Page 23 1.5 Polar curves and infiections......Page 26 2.1 Solution in power series......Page 29 2.2 Convergent power series......Page 33 2.3 Curves, branches, multiplicities and tangents......Page 41 2.4 Factorisation......Page 45 2.5 Notes......Page 48 2.6 Exercises......Page 51 3.1 Puiseux characteristics......Page 53 3.2 Blowing up......Page 54 3.3 Resolution of singularities......Page 56 3.4 Geometry of the resolution......Page 60 3.5 Infinitely near points......Page 63 3.6 The dual graph......Page 71 3.7 Notes......Page 77 3.8 Exercises......Page 79 4.1 Exponents of contact and intersection numbers......Page 81 4.2 The Eggers tree......Page 89 4.3 The semigroup of a branch......Page 92 4.4 Intersections and infinitely near points......Page 102 4.5 Decomposition of transverse polar curves......Page 105 4.6 Notes......Page 108 4.7 Exercises......Page 113 5.1 Vector fields......Page 117 5.2 Knots and links......Page 122 5.3 Description of the geometry of the link......Page 125 5.4 Cable knots......Page 130 5.5 The Alexander polynomial......Page 136 5.6 Notes......Page 142 5.7 Exercises......Page 143 6.1 Fibrations......Page 145 6.2 The Milnor fibration......Page 147 6.3 First properties of the Milnor fibre......Page 153 6.4 Euler characteristics and fibrations......Page 155 6.5 Further formulae for μ......Page 158 6.6 Notes......Page 165 6.7 Exercises......Page 167 7.1 The genus of a singular curve......Page 170 7.2 The degree of the dual curve......Page 173 7.3 Constructible functions and Klein’s equation......Page 176 7.4 The singularities of the dual......Page 184 7.5 Singularities of curves of a given degree......Page 189 7.6 Notes......Page 195 7.7 Exercises......Page 198 8.1 The homology of a blow-up......Page 201 8.2 The exceptional divisor of a curve......Page 208 8.3 Functions on the tree......Page 212 8.4 The topological zeta function......Page 216 8.5 Calculations for a single branch......Page 223 8.6 Notes......Page 230 8.7 Exercises......Page 231 9 Decomposition of the link complement and the Milnor fibre......Page 233 9.1 Canonical decomposition theorems......Page 234 9.2 The complement of an algebraic link......Page 238 9.3 Resolution and plumbing......Page 241 9.4 The Eggers tree and the resolution tree......Page 251 9.5 Finiteness of the monodromy......Page 256 9.6 Seifert fibre spaces......Page 257 9.7 The Eisenbud–Neumann diagram......Page 261 9.8 Calculation of E–N diagrams......Page 265 9.9 The polar discriminant......Page 272 9.10 Notes......Page 275 9.11 Exercises......Page 277 10 The monodromy and the Seifert form......Page 279 10.1 De.nition of Seifert forms......Page 280 10.2 Use of the Thurston decomposition......Page 283 10.3 Calculation of the monodromy......Page 286 10.4 Algebraic classification of Seifert forms......Page 297 10.5 Hermitian forms......Page 303 10.6 Signatures......Page 311 10.7 Proof of 10.6.2 and 10.6.3......Page 316 10.8 Notes......Page 325 10.9 Exercises......Page 328 11.1 Blowing up ideals......Page 331 11.2 The valuative closure of an ideal......Page 337 11.3 Ideals and clusters......Page 342 11.4 Integrally closed ideals......Page 349 11.5 Jets and determinacy......Page 355 11.6 Local rings and di.erentials......Page 362 11.7 Notes......Page 366 11.8 Exercises......Page 368 References......Page 371 Index......Page 382 This book has arisen from the author's successful course at Liverpool University. The text covers all the essentials in a style that is detailed and expertly written by one of the foremost researchers and teachers working in the field. Ideal for either course use or independent study, the volume guides students through the key concepts that will enable them to move on to more detailed study or research within the field. The study of singularities uses techniques from algebra, algebraic geometry, complex analysis and topology. This book introduces graduate students to this attractive area of mathematics. It is based on a MSc course taught by the author and also is an original synthesis, with new views and results not found elsewhere Singularities of plane curves are best understood in terms of complex numbers. This text introduces readers to an attractive area of mathematics that uses techniques from algebra, algebraic geometry, complex analysis & topology There are many ways to give a precise interpretation of the informal notion of a plane curve as something drawn on a sheet of paper.