  Koll\'ar proved that a very general $n$-dimensional complex hypersurface of
degree at least $3\lceil (n+3)/4\rceil$ is not birational to a fibration in
rational curves. This is most interesting when the hypersurface is Fano, in
which case it is covered by rational curves. In this paper, we extend
Koll\'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 to
fibrations in genus $g$ curves. In other words, we show that the fibering genus
of these hypersurfaces can be arbitrarily large. The fibering genus of a
variety has been studied in work of Konno, Ein--Lazarsfeld, and Voisin, but
this is the first paper to explore these ideas in the Fano range. Following
Koll\'ar, we degenerate to characteristic $p>0$ to rule out these fibrations. A
crucial input is Tate's genus change formula and its generalizations, which
imply that any regular curve of genus $g$ is smooth if $p$ is sufficiently
large compared to $g$.
