Restrictions on rational surfaces lying in very general hypersurfaces

We study rational surfaces on very general Fano hypersurfaces in Formula Presented, with an eye toward unirationality. We prove that given any fixed family of rational surfaces, a very general hypersurface of degree d sufficiently close to n and n sufficiently large will admit no maps from surfaces in that family. In particular, this shows that for such hypersurfaces, any rational curve in the space of rational curves must meet the boundary. We also prove that for any fixed ratio Formula Presented, a very general hypersurface in Formula Presented of degree d sufficiently close to n will admit no generically finite maps from a surface satisfying Formula Presented, where H is the pullback of the hyperplane class from Formula Presented and K is the canonical bundle on the surface.