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The space of degree d smooth projective hypersurfaces of CP n admits ascanning map to a certain space of sections. We compute a rational homotopymodel of the action by conjugation of the group U (n + 1) on this space ofsections, from which we deduce that the scanning map induces a monomorphism oncohomology when d > 2. Our main technique is the rational homotopy theory ofSullivan and, more specifically, a Sullivan model for the action by conjugationof a connected topological group on a space of sections.

On the scanning map and the space of smooth projective hypersurfaces