# Algebraic Geometry Seminar

#### The fibering genus of Fano hypersurfaces

**Speaker:**
Nathan Chen, Columbia University

**Location:**
Warren Weaver Hall 512

**Date:**
Tuesday, September 26, 2023, 3:30 p.m.

**Synopsis:**

Kollár proved that a very general n-dimensional complex hypersurface of degree at least 3⌈(n + 3)/4⌉ 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 Kollár’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. Our methods involve degenerating to positive characteristic, and a key input is a generalization of Tate’s genus change formula. This is joint work with Ben Church, Lena Ji, and David Stapleton.