Enumerative geometry of Legendrian foliations: a Tale of Contact

A contact distribution on projective three-space is defined by the 1-form $x_{2}d{x}_1-x_{1}d{x}_2+x_{4}d{x}_3-x_{3}d{x}_4$, up to a change of projective coordinates. The family of contact distributions is parameterized by the complement of the Pfaff-Plücker quadric in the projective 5-space of antisymmetric $4\times 4$ matrices. A foliation of dimension 1 and degree $d$ is specified by a polynomial vector field $\sum p_i{\mathrm{\partial }}_{x_i},p_i$ homogeneous of degree $d$. The foliation is called Legendrian if tangent to some distribution of contact. Our goal is to give formulas for the dimensions and degrees of the varieties of Legendrian foliations, and of the varieties of foliations tangent to a pencil of planes.