Fano hypersurfaces with arbitrarily large degrees of irrationality (N. Chen & D. Stapleton)

In 1995 Kollar proved that a very general complex Fano hypersurface of sufficiently low Fano index is not ruled. His construction involved specialization to positive characteristic due to Mori, in which case the result can be obtained, and Matsusaka's theorem stating that ruledness is a closed property in good families. The authors proceed to prove that for a very general hypersurface it is possible to give a lower bound (growing with dimension) for a degree of the rational dominant map to a ruled variety, in particular, to a rational variety. The previous lower bound of irrationality degree was 3 due to the argument that super-rigid variety does not admit covers of degree 2. We will resketch Kollar's construction and follow the proofs of both statements.