Profiles, linear spaces, and unirationality of complete intersections

Complete intersections may be unexpectedly simple over fields of positive characteristic: for instance, they may be unirational despite being of general type. One explanation is given by profiles, structure that tracks the special shape of polynomials, refining the degree. The aim of this work is to show that complete intersections with small profile should be considered simple by generalizing two classical results on low degree complete intersections: First, the basic geometry of Fano schemes associated with complete intersections depends only on the profile, so that complete intersections with small profile contain many linear spaces. Second, a general complete intersection is unirational once its dimension is sufficiently large compared to its profile.