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 $F$ is determined by the zero locus of $F$. 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

On the height of foliated surfaces with vanishing Kodaira dimension

We prove that the height of a foliated surface of Kodaira dimension zero belongs to {1, 2, 3, 4, 5, 6, 8, 10, 12}. We also construct an explicit projective model for Brunella's very special foliation

Closed meromorphic 1-forms

We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities under the lenses of the theory of closed meromorphic $1$-forms and flat meromorphic connections. We apply the theory to investigate the algebraicity separatrices in a semi-global setting (neighborhood of a compact curve contained in the singular set of the foliation), and the geometry of smooth hypersurfaces with numerically trivial normal bundle on compact K\"ahler manifolds

Vector Fields, Invariant Varieties and Linear Systems

We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criteria for the existence of rational first integrals of a given degree, bounds for the number of first integrals on families of vector fields and a generalization of Darboux's criteria. We also provide a new proof of Gomez-Mont's result on foliations with all leaves algebraic.Comment: 15 pages, Late

Codimension one foliations in positive characteristic

We investigate the geometry of codimension one foliations on smooth projective varieties defined over fields of positive characteristic with an eye toward applications to the structure of codimension one holomorphic foliations on projective manifolds