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Intersections of 1-forms and valuations in a local regular surface. (English) Zbl 0832.14001

The author introduces intersection multiplicities between singular differential 1-forms in the plane and valuations centered at the plane. He proves a “Noether formula” showing the behaviour of the intersection multiplicity under quadratic transformations. Several methods to compute this number are derived. In the last part of the paper, intersection numbers are used to give a description of the desingularization of a 1-form without performing blow-ups.

MSC:

14B05 Singularities in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
32S65 Singularities of holomorphic vector fields and foliations
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