On the scanning map and the space of smooth complex projective hypersurfaces

Abstract The space of degree d smooth projective hypersurfaces of $\mathbb{C}P^n$ admits a scanning map to a certain space of sections. We compute a rational homotopy model of the action by conjugation of the group $U(n+1)$ on this space of sections, from which we deduce that the scanning map induces a monomorphism on cohomology when d > 2. Our main technique is the rational homotopy theory of Sullivan and, more specifically, a Sullivan model for the action by conjugation of a connected topological group on a space of sections.