We describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety.
We study codimension one holomorphic distributions on projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2. We show how the connectedness of the curves in the singular sets of foliations is an integrable phenomenon. This part of the talk is work joint with M. Jardim(Unicamp) and O. Calvo-Andrade(Cimat).
We also study foliations by curves via the investigation of their singular schemes and conormal sheaves and we provide a classification of foliations of degree at most 3 with conormal sheaves locally free. Foliations of degrees 1 and 2 are aways given by a global intersection of two codimension one distributions. In the classification of degree 3 appear Legendrian foliations, foliations whose conormal sheaves are instantons and other ” exceptional”
type examples. This part of the talk is work joint with M. Jardim(Unicamp) and S. Marchesi(Unicamp).