Geodesic
FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY
Forum of Mathematics, Sigma, Tome 8 (2020)

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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. 
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     author = {NATHAN CHEN and DAVID STAPLETON},
     title = {FANO {HYPERSURFACES} {WITH} {ARBITRARILY} {LARGE} {DEGREES} {OF} {IRRATIONALITY}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
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     year = {2020},
     doi = {10.1017/fms.2020.20},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2020.20/}
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NATHAN CHEN; DAVID STAPLETON. FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.20

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