In this work we study codimension one holomorphic distributions on$mathbb{P}^3$. We analyze the properties of their singular schemes and thesemistability of their tangent sheaves. As consequence we give a classificationof locally free distributions of degree $leq 2$ and we show that codimensionone distributions with only isolated singularities have stable tangent sheaves.We describe the moduli spaces of distributions in terms of the Grothendieck'sQuot-scheme for tangent bundles. In some cases we show that the moduli spacesof distributions are irreducible, nonsingular quasi-projective varieties andhave structure of a fibration by projective spaces over a rational base.Finally, in appendix we prove a Bertini type Theorem for reflexive sheaves toconstruct distributions and to provide a description of spaces of distributionsin terms of moduli spaces of stable reflexive sheaves.
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