FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY
We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$, then the degree of irrationality of a very general complex Fano hypersurface of index $e$ and dimension n is bounded from below by a constant times $\sqrt{n}$. To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.