For fixed Fano index e, a very general complex Fano hypersurface of dimension n has degree of irrationality at least sqrt(n)/4 for all sufficiently large n; this is the first construction of rationally connected varieties with degree of irrationality at least 4.
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Fano hypersurfaces with arbitrarily large degrees of irrationality
For fixed Fano index e, a very general complex Fano hypersurface of dimension n has degree of irrationality at least sqrt(n)/4 for all sufficiently large n; this is the first construction of rationally connected varieties with degree of irrationality at least 4.