Fano hypersurfaces and their birational geometry

TL;DR

This paper surveys recent results on the nonrationality and birational rigidity of Fano hypersurfaces, emphasizing those of index one and discussing higher index cases to understand their birational properties.

Contribution

It provides a comprehensive overview of the current state of knowledge on the birational geometry of Fano hypersurfaces, highlighting key results and open questions.

Findings

Fano hypersurfaces of index one exhibit nonrationality and birational rigidity.

Higher index Fano hypersurfaces are also analyzed for their birational properties.

The survey summarizes significant progress and remaining challenges in the field.

Abstract

We survey some results on the nonrationality and birational rigidity of certain hypersurfaces of Fano type. The focus is on hypersurfaces of Fano index one, but hypersurfaces of higher index are also discussed.