Resonance webs of hyperplane arrangements

Each irreducible component of the first resonance variety of a hyperplane arrangement naturally determines a codimension one foliation on the ambient space. The superposition of these foliations define what we call the resonance web of the arrangement. In this paper we initiate the study of these objects with emphasis on their spaces of abelian relations.Comment: (v2) Minor changes following suggestions of the referee. To appear in the Proceedings of the 2nd MSJ-SI on Arrangements of Hyperplane

An analogous of Jouanolou's Theorem in positive characteristic

We show that a generic vector field on an affine space of positive characteristic admits an invariant algebraic hypersurface. This contrast with Jouanolou's Theorem that shows that in characteristic zero the situation is completely opposite. That is a generic vector field in the complex plane does not admit any invariant algebraic curve.Comment: 5 pages, LaTe

Foliations invariant by rational maps

We give a classification of pairs (F, f) where F is a holomorphic foliation on a projective surface and f is a non-invertible dominant rational map preserving F. We prove that both the map and the foliation are integrable in a suitable sense.Comment: 17 pages. To appear in Math. Zeitshrift