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We study codimension one holomorphic distributions on the projectivethree-space, analyzing the properties of their singular schemes and tangentsheaves. In particular, we provide a classification of codimension onedistributions of degree at most 2 with locally free tangent sheaves, and showthat codimension one distributions of arbitrary degree with only isolatedsingularities have stable tangent sheaves. Furthermore, we describe the modulispace of distributions in terms of Grothendieck's Quot-scheme for the tangentbundle. In certain cases, we show that the moduli space of codimension onedistributions on the projective space is an irreducible, nonsingularquasi-projective variety. Finally, we prove that every rational foliation, andcertain logarithmic foliations have stable tangent sheaves.

Codimension one holomorphic distributions on the projective three-space