The fibering genus of Fano hypersurfaces

TL;DR

This paper demonstrates that Fano hypersurfaces can have arbitrarily large fibering genus, extending Kollár's work and using degeneration techniques in positive characteristic to analyze fibrations in genus g curves.

Contribution

It shows that for any genus g, there exist Fano hypersurfaces not birational to fibrations in genus g curves, revealing the unbounded nature of fibering genus in this class.

Findings

Fano hypersurfaces can have arbitrarily large fibering genus.

Degeneration to characteristic p>0 is used to analyze fibrations.

Tate's genus change formula helps determine smoothness of curves.

Abstract

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 $g$.