An h-principle for complements of discriminants

We compare spaces of nonsingular algebraic sections of ample vector bundles to spaces of continuous sections of jet bundles. Under some conditions, we provide an isomorphism in homology in a range of degrees growing with the jet ampleness. As an application, when L is a very ample line bundle on a smooth projective complex variety, we prove that the rational cohomology of the space of nonsingular algebraic sections of L^⊗d stabilises as d → ∞ and compute the stable cohomology. We also prove that the integral homology does not stabilise, using tools from stable homotopy theory.