  We study the locus of smooth hypersurfaces inside the Hilbert scheme of a
smooth projective complex variety. In the spirit of scanning, we construct a
map to a continuous section space of a projective bundle, and show that it
induces an isomorphism in integral homology in a range of degrees growing with
the ampleness of the hypersurfaces. When the ambient variety is a curve, this
recovers a result of McDuff about configuration spaces. We compute the rational
cohomology of the section space and exhibit a phenomenon of homological
stability for hypersurfaces with first Chern class going to infinity. For
simply connected varieties, the rational cohomology is shown to agree with the
stable cohomology of a moduli space of hypersurfaces, with a peculiar
tangential structure, as studied by Galatius and Randal-Williams.
