##
**Singular points of plane curves.**
*(English)*
Zbl 1057.14001

London Mathematical Society Student Texts 63. Cambridge: Cambridge University Press (ISBN 0-521-83904-1/hbk; 0-521-54774-1/pbk). xi, 370 p. (2004).

The study of singular points of algebraic curves in the complex plane has a long history. Its beginnings can be traced back to Sir Isaac Newton, and the algebraic geometers of the nineteenth and early twentieth century developed it into a fascinating, already remarkably rich theory. One of the major achievements, during this period, was the resolution of singularities of such curves initiated by Max Noether.

From the 1920s on, the then new topological methods were applied to the local study of singularities of curves, knots and links. In the second half of the twentieth century, the newly developing singularity theory in higher dimensions also propelled the study of the singular points of plane curves, and the developments in this area have been tremendous since the late 1960s. In the course of its long history, singularity theory of plane curves has grown into a meeting point for many different disciplines of mathematics, including algebra, complex analysis, algebraic geometry, topology, and combinatorics. The interaction between ideas, methods and techniques from these various sources makes the study of singularities of plane curves particularly fascinating, enlightening, abundant and fruitful. Moreover, this subject provides a beautiful testing ground for geometric ideas, in general, and a perfect topic for developing a profound understanding of the principles of modern geometry, likewise.

The book under review, written by one of the leading experts in singularity theory, is a highly welcome attempt to present a systematic, comprehensive, versatile and up-to-date account of the present state of art of this venerable area within mathematics. Based on an M.Sc. course taught a number of times (since 1975) at the University of Liverpool, it has partly the character of an introductory textbook, and can be used as such, but it also discusses more recent, advanced and intradisciplinary topics from the forefront of current research in the singularity theory of plane curves. Thus the text, consisting of eleven chapters, is virtually divided into two main parts.

The first five chapters are kept to the level of the underlying M.Sc. course and, therefore, are more introductory and elementary in nature. They are meant to form the core of the book, providing the foundations of the classical theory of plane curve singularities. As for this part, the author has chosen the concept of equisingularity, i.e., the most important equivalence relation for singularities, as the general leitmotif for his approach. Equisingularity can be characterized from numerous different points of view, and the development of the distinct ideas and methods leading to that same concept is taken as the frame for an introduction to curve singularities. This is the mean feature of this approach, and of the book as a whole, that the author emphasizes the equivalence of differing concepts and methods from the beginning on, thereby demonstrating their appearance and power in an integrated account.

Chapter 1 compiles the necessary preliminary material: the definition of algebraic curves in the plane, intersection numbers, resultants and discriminants, manifolds and the implicit function theorem, polar curves and inflection points. All this is treated as basically familiar background material and not covered in every detail. The story starts with Chapter 2, where parametrizations of curves via Puiseux power series, branches of curves, multiplicities and tangent lines to curves are discussed. This is used in Chapter 3 to describe the resolution of curve singularities, including the blow-up process, the notion of infinitely near points, invariants of singularities, and the graph-theoretic interpretation of the configurations arising in the resolution process. Chapter 4 deals with the theory of contact of two branches of a curve, the Eggers tree associated with a branch, computing intersection numbers for curves with several branches, and the equivalent characterizations of the concept of equisingularity in the whole framework developed so far. Chapter 5 turns to the topological aspects of curve singularities, with a special emphasis on knots, links and the classical Alexander polynomial. Equisingularity is then reconsidered from this topological point of view.

The second part of the book, which comprises the remaining six chapters, is written at a more sophisticated level, gives introductions to a number of topics of current research, and even offers several new results of the author. In these more advanced chapters, the topological aspects of curve singularities play a dominant role.

Chapter 6 is devoted to the Milnor fibration, Milnor numbers, and the Euler characteristic of a fibration. The latter is used for several instructive calculations of Milnor numbers. Chapter 7 is entitled “Projective curves and their duals”. The author gives proofs of the general Plücker theorems for singular plane curves, treats Klein’s equation by using Euler characteristics of constructible functions, analyzes the singularities of a dual curve, and surveys some known results about curves with so-called maximal singularities.

The following three chapters are very up-to-date and lead up to the calculation of the monodromy of the Milnor fibration. Chapter 8 introduces calculations and notation for later use, including several numerical invariants of singularities and their representation using exceptional cycles on resolution trees. This chapter also contains an introduction to the topological zeta function à la Denef-Loeser.

Chapter 9 discusses the application of W. Thurston’s decomposition theorems for 3-manifolds and for homeomorphisms of surfaces to the Milnor fibration. This chapter offers a novel view to the topology of curve singularities, with a number of results published for the first time. Among other things, the author presents a finiteness criterion for the monodromy and a close relation between the Eggers tree, the resolution graph and the Eisenbud-Neumann diagram of a singularity. Chapter 10 continues with new results in calculating the monodromy, mainly by using Seifert matrices and (again), the Thurston decomposition theorems. In addition, it is shown how to classify Seifert forms over a field, and how some of the numerical invariants required for the classification over the rational number field can be calculated.

The final Chapter 11 touches upon the more algebraic aspects of curve singularities. Ideals in the local ring of a singularity are related to the exceptional cycles studied earlier, and a link between ideals and Enriques’s clusters of infinitely near points is established in form of a Galois correspondence. The discussion concludes with brief treatments of jets, equisingularity classes and the determinacy of functions.

Each chapter of the book comes with a section on “Notes” and “Exercises”. The notes include historical remarks, references, comments on related material not covered in the text, and some hints for further research. The exercises form a balanced mixture of routine problems on applying the results in the text to concrete examples, on the one hand, and more challenging problems related to an alternative approach to a topic treated before, on the other. Also, the entire text is relaxed by numerous illustrating and instructive examples, and the bibliography is with more than 200 references more than ample.

All in all, this book, being partly an introductory textbook and partly an advanced research monograph, is extremely comprehensive, profuse and versatile. It contains a wealth of information, both classical and topical, on the attractive and evergreen area of plane algebraic curves, and it offers a lot of new insights to all kinds of readers.

The text reflects the author’s great expertise in the field in a masterly way, and that just as much as his passion for the subject and his cultured attitude. His style of writing mathematics is utmost pleasant, nowhere formal, very user-friendly, throughout motivating and highly inspiring. No doubt, this book will quickly become a widely used standard text on singularities of plane curves, and a valuable reference book, too.

From the 1920s on, the then new topological methods were applied to the local study of singularities of curves, knots and links. In the second half of the twentieth century, the newly developing singularity theory in higher dimensions also propelled the study of the singular points of plane curves, and the developments in this area have been tremendous since the late 1960s. In the course of its long history, singularity theory of plane curves has grown into a meeting point for many different disciplines of mathematics, including algebra, complex analysis, algebraic geometry, topology, and combinatorics. The interaction between ideas, methods and techniques from these various sources makes the study of singularities of plane curves particularly fascinating, enlightening, abundant and fruitful. Moreover, this subject provides a beautiful testing ground for geometric ideas, in general, and a perfect topic for developing a profound understanding of the principles of modern geometry, likewise.

The book under review, written by one of the leading experts in singularity theory, is a highly welcome attempt to present a systematic, comprehensive, versatile and up-to-date account of the present state of art of this venerable area within mathematics. Based on an M.Sc. course taught a number of times (since 1975) at the University of Liverpool, it has partly the character of an introductory textbook, and can be used as such, but it also discusses more recent, advanced and intradisciplinary topics from the forefront of current research in the singularity theory of plane curves. Thus the text, consisting of eleven chapters, is virtually divided into two main parts.

The first five chapters are kept to the level of the underlying M.Sc. course and, therefore, are more introductory and elementary in nature. They are meant to form the core of the book, providing the foundations of the classical theory of plane curve singularities. As for this part, the author has chosen the concept of equisingularity, i.e., the most important equivalence relation for singularities, as the general leitmotif for his approach. Equisingularity can be characterized from numerous different points of view, and the development of the distinct ideas and methods leading to that same concept is taken as the frame for an introduction to curve singularities. This is the mean feature of this approach, and of the book as a whole, that the author emphasizes the equivalence of differing concepts and methods from the beginning on, thereby demonstrating their appearance and power in an integrated account.

Chapter 1 compiles the necessary preliminary material: the definition of algebraic curves in the plane, intersection numbers, resultants and discriminants, manifolds and the implicit function theorem, polar curves and inflection points. All this is treated as basically familiar background material and not covered in every detail. The story starts with Chapter 2, where parametrizations of curves via Puiseux power series, branches of curves, multiplicities and tangent lines to curves are discussed. This is used in Chapter 3 to describe the resolution of curve singularities, including the blow-up process, the notion of infinitely near points, invariants of singularities, and the graph-theoretic interpretation of the configurations arising in the resolution process. Chapter 4 deals with the theory of contact of two branches of a curve, the Eggers tree associated with a branch, computing intersection numbers for curves with several branches, and the equivalent characterizations of the concept of equisingularity in the whole framework developed so far. Chapter 5 turns to the topological aspects of curve singularities, with a special emphasis on knots, links and the classical Alexander polynomial. Equisingularity is then reconsidered from this topological point of view.

The second part of the book, which comprises the remaining six chapters, is written at a more sophisticated level, gives introductions to a number of topics of current research, and even offers several new results of the author. In these more advanced chapters, the topological aspects of curve singularities play a dominant role.

Chapter 6 is devoted to the Milnor fibration, Milnor numbers, and the Euler characteristic of a fibration. The latter is used for several instructive calculations of Milnor numbers. Chapter 7 is entitled “Projective curves and their duals”. The author gives proofs of the general Plücker theorems for singular plane curves, treats Klein’s equation by using Euler characteristics of constructible functions, analyzes the singularities of a dual curve, and surveys some known results about curves with so-called maximal singularities.

The following three chapters are very up-to-date and lead up to the calculation of the monodromy of the Milnor fibration. Chapter 8 introduces calculations and notation for later use, including several numerical invariants of singularities and their representation using exceptional cycles on resolution trees. This chapter also contains an introduction to the topological zeta function à la Denef-Loeser.

Chapter 9 discusses the application of W. Thurston’s decomposition theorems for 3-manifolds and for homeomorphisms of surfaces to the Milnor fibration. This chapter offers a novel view to the topology of curve singularities, with a number of results published for the first time. Among other things, the author presents a finiteness criterion for the monodromy and a close relation between the Eggers tree, the resolution graph and the Eisenbud-Neumann diagram of a singularity. Chapter 10 continues with new results in calculating the monodromy, mainly by using Seifert matrices and (again), the Thurston decomposition theorems. In addition, it is shown how to classify Seifert forms over a field, and how some of the numerical invariants required for the classification over the rational number field can be calculated.

The final Chapter 11 touches upon the more algebraic aspects of curve singularities. Ideals in the local ring of a singularity are related to the exceptional cycles studied earlier, and a link between ideals and Enriques’s clusters of infinitely near points is established in form of a Galois correspondence. The discussion concludes with brief treatments of jets, equisingularity classes and the determinacy of functions.

Each chapter of the book comes with a section on “Notes” and “Exercises”. The notes include historical remarks, references, comments on related material not covered in the text, and some hints for further research. The exercises form a balanced mixture of routine problems on applying the results in the text to concrete examples, on the one hand, and more challenging problems related to an alternative approach to a topic treated before, on the other. Also, the entire text is relaxed by numerous illustrating and instructive examples, and the bibliography is with more than 200 references more than ample.

All in all, this book, being partly an introductory textbook and partly an advanced research monograph, is extremely comprehensive, profuse and versatile. It contains a wealth of information, both classical and topical, on the attractive and evergreen area of plane algebraic curves, and it offers a lot of new insights to all kinds of readers.

The text reflects the author’s great expertise in the field in a masterly way, and that just as much as his passion for the subject and his cultured attitude. His style of writing mathematics is utmost pleasant, nowhere formal, very user-friendly, throughout motivating and highly inspiring. No doubt, this book will quickly become a widely used standard text on singularities of plane curves, and a valuable reference book, too.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14B05 | Singularities in algebraic geometry |

14H20 | Singularities of curves, local rings |

14H50 | Plane and space curves |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

58K10 | Monodromy on manifolds |