Introduction to singularities and deformations.

*(English)*Zbl 1125.32013
Springer Monographs in Mathematics. Berlin: Springer (ISBN 978-3-540-28380-5/hbk). xii, 471 p. (2007).

The book under review presents the elements of the singularity theory of analytic spaces with applications; it consists of a preface, two main parts, three appendices and a bibliography including 158 items among which are 18 references on works written by the authors with collaborators.

The first part deals with complex spaces and germs. It contains basic notions and results of the general theory such as the Weierstraß preparation theorem, the finite coherence theorem with applications, finite and flat morphisms, normalization, singular locus and relations with differential calculus. In addition, the cases of isolated hypersurface and plane curve singularities are treated. Thus the authors describe some well-known invariants of hypersurface singularities including the Milnor and Tjurina numbers and methods of their computation. They also discuss the concept of finite determinacy, the property of quasihomogeneity, algebraic group actions, the classification of simple singularities, the parameterization and resolution of plane curve singularities, the intersection multiplicity and the semigroup of values associated with a plane curve singularity, the conductor and other classical topological and analytic invariants.

The second part is concerned with local deformation theory of complex space germs. First the authors describe the general deformation theory of isolated singularities of complex spaces. Then the notions of versality, infinitesimal deformations and obstructions are considered in detail. The final section contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the \(\mu\)-constant stratum which is based on properties of deformations of the parametrization. This result is obtained, in fact, as a further development of ideas by J. M. Wahl [Trans. Am. Math. Soc. 193, 143–170 (1974; Zbl 0294.14007)]. Three appendices include a detail description of basic notions and results from sheaf theory, commutative algebra and formal deformation theory.

The book is written in a clear style, almost all key topics are followed by carefully chosen computational examples together with algorithms implemented in the computer algebra system Singular [see G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3. A computer algebra system for polynomial computations. Centre for Computer Algebra, Univ. Kaiserslautern (2005), http://www.singular.uni-kl.de]. Moreover, the exposition contains many non-formal comments, remarks and good exercises illustrated by nice pictures. Without a doubt this book is comprehensible, interesting and useful for graduate students; it is also very valuable for advanced researchers, lecturers, and practicians working in singularity theory, algebraic geometry, complex analysis, commutative algebra, topology, and in other fields of pure mathematics.

The first part deals with complex spaces and germs. It contains basic notions and results of the general theory such as the Weierstraß preparation theorem, the finite coherence theorem with applications, finite and flat morphisms, normalization, singular locus and relations with differential calculus. In addition, the cases of isolated hypersurface and plane curve singularities are treated. Thus the authors describe some well-known invariants of hypersurface singularities including the Milnor and Tjurina numbers and methods of their computation. They also discuss the concept of finite determinacy, the property of quasihomogeneity, algebraic group actions, the classification of simple singularities, the parameterization and resolution of plane curve singularities, the intersection multiplicity and the semigroup of values associated with a plane curve singularity, the conductor and other classical topological and analytic invariants.

The second part is concerned with local deformation theory of complex space germs. First the authors describe the general deformation theory of isolated singularities of complex spaces. Then the notions of versality, infinitesimal deformations and obstructions are considered in detail. The final section contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the \(\mu\)-constant stratum which is based on properties of deformations of the parametrization. This result is obtained, in fact, as a further development of ideas by J. M. Wahl [Trans. Am. Math. Soc. 193, 143–170 (1974; Zbl 0294.14007)]. Three appendices include a detail description of basic notions and results from sheaf theory, commutative algebra and formal deformation theory.

The book is written in a clear style, almost all key topics are followed by carefully chosen computational examples together with algorithms implemented in the computer algebra system Singular [see G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3. A computer algebra system for polynomial computations. Centre for Computer Algebra, Univ. Kaiserslautern (2005), http://www.singular.uni-kl.de]. Moreover, the exposition contains many non-formal comments, remarks and good exercises illustrated by nice pictures. Without a doubt this book is comprehensible, interesting and useful for graduate students; it is also very valuable for advanced researchers, lecturers, and practicians working in singularity theory, algebraic geometry, complex analysis, commutative algebra, topology, and in other fields of pure mathematics.

Reviewer: Aleksandr G. Aleksandrov (Moskva)

##### MSC:

32S30 | Deformations of complex singularities; vanishing cycles |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32S10 | Invariants of analytic local rings |

32S15 | Equisingularity (topological and analytic) |

32S25 | Complex surface and hypersurface singularities |

32S45 | Modifications; resolution of singularities (complex-analytic aspects) |

14B07 | Deformations of singularities |

14J17 | Singularities of surfaces or higher-dimensional varieties |

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

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