  We study the space of smooth marked hypersurfaces in a given linear system.
Specifically, we prove a homology h-principle to compare it with a space of
sections of an appropriate jet bundle. Using rational models, we compute its
rational cohomology in a range of degrees, and deduce a homological stability
result for hypersurfaces of increasing degree. We also describe the Hodge
weights on the stable cohomology, and thereby connect our topological result to
motivic results of Howe.
