Codimension-one foliations on adjoint varieties

In this paper, we classify codimension-one foliations on adjoint varieties with most positive anti-canonical class.

On adjoint varieties of Picard number one, we show that such foliations are always induced by a pencil of hyperplane sections with respect to the minimal embedding.

For adjoint varieties of Picard number two, we prove that the space of such foliations contains more than one irreducible component, and we describe each of them.

As a tool for understanding these foliations, we introduce the concept of the degree of a foliation with respect to a family of rational curves, which may be of independent interest.