12,833 research outputs found
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
Webs invariant by rational maps on surfaces
We prove that under mild hypothesis rational maps on a surface preserving
webs are of Latt\`es type. We classify endomorphisms of P^2 preserving webs,
extending former results of Dabija-Jonsson.Comment: 27 pages, submitte
On the degree of Polar Transformations -- An approach through Logarithmic Foliations
We investigate the degree of the polar transformations associated to a
certain class of multi-valued homogeneous functions. In particular we prove
that the degree of the pre-image of generic linear spaces by a polar
transformation associated to a homogeneous polynomial is determined by the
zero locus of . For zero dimensional-dimensional linear spaces this was
conjecture by Dolgachev and proved by Dimca-Papadima using topological
arguments. Our methods are algebro-geometric and rely on the study of the Gauss
map of naturally associated logarithmic foliations
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