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**Singularities of plane curves.**
*(English)*
Zbl 0967.14018

London Mathematical Society Lecture Note Series. 276. Cambridge: Cambridge University Press. xv, 345 p. (2000).

In this book the author studies properties of germs of plane curves, i.e., power series in \({\mathbb C}[[x, y]]\), the ring of formal power series in two indeterminates over the complex numbers, or power series in \({\mathbb C}\{x, y\}\), the ring of convergent power series. After some introductory remarks, the first chapter deals with Puiseux’s theorem and the Newton-Puiseux algorithm, and the second chapter introduces branches of a germ and the notion of intersection multiplicity of two curves. Blowing up and infinitely near points are the subject of the third chapter; Enrique’s definition of infinitely near points is also mentioned. This chapter deals also with the notion of proximate points, free and satellite points, and the resolution of singularities for germs of curves. There are also algebraic descriptions of the rings in the successive neighborhoods of the ring of a plane germ (cf. section 3.10 and 3.11). The notion of virtual multiplicity of clusters is defined in chapter 4; this chapter contains also a proof of Noether’s \(Af + B\varphi\) theorem. Chapter 5 deals with characteristic exponents, the position of infinitely near points and the semigroup of values of an irreducible germ. In particular, it is shown that two irreducible germs are equisingular (i.e., have the same characteristic exponents) iff they have the same semigroup. (The well-known necessary and sufficient condition for a numerical semigroup to be the semigroup of a plane irreducible germ is given as exercises 5.10 and 5.11.) Polar germs are dealt with in chapter 6. Here the author gives Pham’s example (showing that equisingular curves may have non-equisingular polars), proves Merle’s result on the polar quotients, introduces the Milnor number, and shows [using result of his two papers: Math. Ann. 287, 429-454 (1990; Zbl 0675.14009); Comp. Math. 89, 339-359 (1993; Zbl 0806.14021)] that in the case of a single characteristic exponent, the polars of a germ may be used for unveiling properties of a germ that depend on its isomorphism class and not only on its equisingularity class. In section 6.10 and 6.11 the author treats the polar invariants of a reduced germ \(\xi\) and shows how to compute them from an Enrique diagram of \(\xi\).

Linear families of germs are the subject of chapter 7. In section 7.2 the author introduces the weighted clusters of base points of pencils and proves their main properties. Section 7.5 and 7.6 deals with the notion of \(E\)-sufficiency (or \(C^0\)-sufficiency): A positive integer \(n\) is said to \(E\)-sufficient for a reduced \(f \in {\mathbb C} [[x,y]]\) iff all \(h \in f + m^n\) are non-zero, reduced and equisingular to \(f\) (\(m\) is the maximal ideal of \({\mathbb C} [[x, y]]\)). ({}The letter \(E\) stands for equisingularity: there is a similar notion with regard to being analytically isomorphic which is dealt with in section 7.7.) The last chapter presents – in the context of this book – a classification of valuations of \({\mathbb C}\{x,y\}\) and a proof of Zariski’s factorization theorem for complete ideals in \({\mathbb C}\{x, y\}\).

This book covers a lot of classical material in a modern fashion; in particular, it clarifies some obscure definitions, remarks and statements in older papers from the end of the 19th century and the beginning of the 20th century. As prerequisites for reading this book a course in algebra should be enough. The clear presentation of the proofs makes life for a reader very easy; exercises at the end of each chapter are either examples to give the reader the possibility of proving for himself that he has mastered the text or provide new theorems not dealt with in the main text.

Taken all together: A book making nice reading; it belongs to the shelf of every mathematician interested in curves and their singularities.

Linear families of germs are the subject of chapter 7. In section 7.2 the author introduces the weighted clusters of base points of pencils and proves their main properties. Section 7.5 and 7.6 deals with the notion of \(E\)-sufficiency (or \(C^0\)-sufficiency): A positive integer \(n\) is said to \(E\)-sufficient for a reduced \(f \in {\mathbb C} [[x,y]]\) iff all \(h \in f + m^n\) are non-zero, reduced and equisingular to \(f\) (\(m\) is the maximal ideal of \({\mathbb C} [[x, y]]\)). ({}The letter \(E\) stands for equisingularity: there is a similar notion with regard to being analytically isomorphic which is dealt with in section 7.7.) The last chapter presents – in the context of this book – a classification of valuations of \({\mathbb C}\{x,y\}\) and a proof of Zariski’s factorization theorem for complete ideals in \({\mathbb C}\{x, y\}\).

This book covers a lot of classical material in a modern fashion; in particular, it clarifies some obscure definitions, remarks and statements in older papers from the end of the 19th century and the beginning of the 20th century. As prerequisites for reading this book a course in algebra should be enough. The clear presentation of the proofs makes life for a reader very easy; exercises at the end of each chapter are either examples to give the reader the possibility of proving for himself that he has mastered the text or provide new theorems not dealt with in the main text.

Taken all together: A book making nice reading; it belongs to the shelf of every mathematician interested in curves and their singularities.

Reviewer: K.-H.Kiyek (Paderborn)

### MSC:

14H20 | Singularities of curves, local rings |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

32-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces |

32S15 | Equisingularity (topological and analytic) |

32S05 | Local complex singularities |

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

14H50 | Plane and space curves |

32B10 | Germs of analytic sets, local parametrization |