Degree of the exceptional component of foliations of degree two and codimension one in $${\mathbb {P}}^{3}$$P3

Abstract

The purpose of this work is to obtain the degree of the exceptional component of the space of holomorphic foliations of degree two and codimension one in P 3 . This component is the closure of the orbit of the foliation defined by the differential form1 2 under the natural action of the group of automorphisms of P 3 . Our first task is to unravel a geometric characterization of the pair g, f . This leads us to the construction of a parameter space as an explicit fiber bundle over the variety of complete flags. Using tools from equivariant intersection theory, especially Bott's formula, the degree is expressed as an integral over our parameter space.

Linear Pullback Components of the Space of Codimension One Foliations

“…Here the leaves of a typical foliation are the orbits of a linear action of the affine group in one variable in P 3 . It is shown in [2] that for n > 3, in addition to the components described above there exists a component E n obtained by linear pullback P n P 3 of foliations in E. The degree of E has been found in [14].…”