Bounding solutions of Pfaff equations

Details Bounding solutions of Pfaff equations Bounding solutions of Pfaff equations

2003-04-11

arxiv.org/abs/math/0304148v1

Mathematics

Algebraic Geometry

18 pages; AMSLaTeX

Scientific paper

Let \omega be a Pfaff system of differential forms on a projective space. Let S be its singular locus, and Y a solution of \omega=0. We prove Y\cap S is of codimension at most 1 in Y, just as Jouanolou suspected; he proved this result assuming \omega is completely integrable, and asked if the integrability is, in fact, needed. Furthermore, we prove a lower bound on the Castelnuovo--Mumford regularity of Y\cap S. As in two related articles, we derive upper bounds on numerical invariants of Y, thus contributing to the solution of the Poincare problem. We work with Pfaff fields not necessarily induced by Pfaff systems, with ambient spaces more general than projective spaces, and usually in arbitrary characteristic.

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