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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.
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