Desingularization Strategies for Three-Dimensional Vector Fields
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1259 Felipe Cano Torres
Desingularization Strategies for Three-Dimensional Vector Fields
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author Felipe Cano Torres Departamento de Algebra y Geometrfa Facultad de Ciencias Valladolid 47005, Spain
This volume is being published in a parallel edition by the Instituto de Maternatica Pura e Aplicada, Rio de Janeiro as volume 43 of the series "Monograffas de Matematica" . Mathematics Subject Classification (1980): 24B05, 32B30, 58A30, 58F 14 ISBN 3-540-17944-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17944-5 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
To
Mercedes
INTRODUCTION
Let 0 = aa/ax + ba/ay
be a plane vector field (or a derivation of a two-
dimensional ring of power series over a field k). We can measure how "bad" is the singularity of 0 at the origin by means of the number v(O) = min(v(a),v(b)), where v(a}, resp. V(b), are the orders at the origin of a, resp. b. Assume moreover that a and b have no common factor (this allows us to consider the "saturated foliation", see 161). In this situation, after making a finite number of quadratic blowing-ups of the ambient space, we obtain v(O)21 at every singular point (see,e.g., Seidenberg's result 1131). The behaviour of the measure v(O) after applying a quadratic blowing-up is not as good as one could expect. For instance, if 0
3 ya/ax + x a/ ay
we have v(O)=1,
but the strict transform in a suitable chart is given by 0'= y'x'a/ax'+(x,2_y,2)a/ay' and the order v(O') at the origin has peen increased by a unit (in general, under any sequence of quadratic blowing-ups the order remains 2 v(O)+1). This difficulty may be avoided by considering the vector field as a vector field which is " tangent to the exceptional divisor x'=O ". In this case a basis of the free module of such vector fields is given by are
y' and
x'a/ax'
and
a/ay', thus the corresponding coefficients for 0'
x,2_ y,2. Now, the adapted order (the minimum of the orders of these
coefficients) is one and it has not been increased. If we take this approach, a similar result can be proved: after a finite number of quadratic blowing-ups we obtain that the adapted order is =OJ
4
and that 0X,P is a U.F.D. because it is regular.
(1.2.4)
Definition. We shall say that an unidimensional distribution
plicati vely
irreducible"
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