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Koll\'ar proved that a very general $n$-dimensional complex hypersurface ofdegree at least $3\lceil (n+3)/4\rceil$ is not birational to a fibration inrational curves. This is most interesting when the hypersurface is Fano, inwhich case it is covered by rational curves. In this paper, we extendKoll\'ar's ideas and show that for any genus $g$, there are Fano hypersurfaces(in more restrictive degree and dimension ranges) that are not birational tofibrations in genus $g$ curves. In other words, we show that the fibering genusof these hypersurfaces can be arbitrarily large. The fibering genus of avariety has been studied in work of Konno, Ein--Lazarsfeld, and Voisin, butthis is the first paper to explore these ideas in the Fano range. FollowingKoll\'ar, we degenerate to characteristic $p>0$ to rule out these fibrations. Acrucial input is Tate's genus change formula and its generalizations, whichimply that any regular curve of genus $g$ is smooth if $p$ is sufficientlylarge compared to $g$.

The fibering genus of Fano hypersurfaces