In this thesis, I prove a general h-principle for algebraic sections of vector bundles, and use it to investigate the homology of moduli spaces of smooth algebraic hypersurfaces. The thesis consists of an introduction followed by three papers, the last of which is joint with Ronno Das.
In the first paper, I consider spaces of algebraic sections of vector bundles subject to differential relations. On smooth projective complex varieties, I prove that the homology of such a space coincides in a range with that of a space of continuous sections of an associated bundle. As an immediate consequence, I show stability of the rational cohomology for complement of discriminants in linear systems of hypersurfaces of increasing degree. This paper is the most technical and its results are used repeatedly throughout the thesis.
In the second paper, I study the locus of smooth hypersurfaces inside the Hilbert scheme of a smooth complex projective variety. Using the results of the first paper, I show how part of its cohomology can be computed via an h-principle akin to a scanning map. I also explain how to compare the rational cohomology to that of classifying spaces of diffeomorphisms groups of hypersurfaces.
In the third paper, Ronno Das and I study the cohomology of the universal smooth hypersurface bundle with marked points. We adapt the arguments of the first paper to show another h-principle. Using rational models, we deduce rational homological stability for this space.