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**Desingularization strategies for three-dimensional vector fields.**
*(English)*
Zbl 0686.14008

Lecture Notes in Mathematics, 1259. Berlin etc.: Springer-Verlag. IX, 194 p.; DM 35.00 (1987).

If D is a holomorphic vector field on a nonsingular algebraic variety V, and \(\pi: W\to V\) is a blowing-up, there is a natural lift of D, or at least of the \({\mathcal O}_ V\)-module generated by D, to W. The desingularisation problem is that of proving the existence of a sequence of blow-ups such that the singularities of the lifted vector field are of a simple kind. For vector fields \(D=a\partial /\partial x+b\partial /\partial y\) on \(K^ 2\), the order \(\nu (D)=\min (ord(a),ord(b))\) may increase on blowing-up, so this is modified by considering vector fields tangent to the exceptional divisor to an ‘adapted order’. This monograph is devoted to the proof that in the 3-dimensional case (over an algebraically closed field of characteristic 0) one can define a sequence of blowings-up to reduce the adapted order to 1.

The first chapter contains general hypotheses and definitions and includes the statement of the theorem. It is necessary at each stage to choose a permissible centre of blowing-up (a regular closed subscheme having normal crossings with E, such that the adapted order cannot increase on blowing-up) so that one has sufficient control of the resulting germs at all centres on the new exceptional locus: the problem can thus be viewed as a 2-person game. It is not difficult to express these orders, exceptional sets, etc. explicitly in terms of suitable local coordinates.

The remainder of the book constructs a winning strategy. The problem is divided into 4 main cases, then into a couple of dozen subcases; the strategy is first devised in the simplest case, then later cases are reduced to previous ones. It is not easy to discern any over-riding pattern, though various techniques from the resolution of singularities of varieties are used.

The first chapter contains general hypotheses and definitions and includes the statement of the theorem. It is necessary at each stage to choose a permissible centre of blowing-up (a regular closed subscheme having normal crossings with E, such that the adapted order cannot increase on blowing-up) so that one has sufficient control of the resulting germs at all centres on the new exceptional locus: the problem can thus be viewed as a 2-person game. It is not difficult to express these orders, exceptional sets, etc. explicitly in terms of suitable local coordinates.

The remainder of the book constructs a winning strategy. The problem is divided into 4 main cases, then into a couple of dozen subcases; the strategy is first devised in the simplest case, then later cases are reduced to previous ones. It is not easy to discern any over-riding pattern, though various techniques from the resolution of singularities of varieties are used.

Reviewer: C.T.C.Wall

### MSC:

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

34M99 | Ordinary differential equations in the complex domain |

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

91A05 | 2-person games |

14B05 | Singularities in algebraic geometry |

14J30 | \(3\)-folds |