The degree of irrationality of hypersurfaces in various Fano varieties

TL;DR

This paper computes the degree of irrationality for high-degree hypersurfaces in various Fano varieties, extending existing techniques and analyzing the geometry of low-degree rational maps and curves within these varieties.

Contribution

It introduces new methods to determine the degree of irrationality for hypersurfaces in diverse Fano varieties, expanding previous work beyond projective space.

Findings

Degree of irrationality computed for hypersurfaces in multiple Fano varieties

Low-degree rational maps' fibers are contained in low-degree curves in Fano varieties

Geometric analysis of curves in Fano varieties informs irrationality measures

Abstract

The purpose of this paper is to compute the degree of irrationality of hypersurfaces of sufficiently high degree in various Fano varieties: quadrics, Grassmannians, products of projective space, cubic threefolds, cubic fourfolds, and complete intersection threefolds of type (2,2). This extends the techniques of Bastianelli, De Poi, Ein, Lazarsfeld, and the second author who computed the degree of irrationality of hypersurfaces of sufficiently high degree in projective space. A theme in the paper is that the fibers of low degree rational maps from the hypersurfaces to projective space tend to lie on curves of low degree contained in the Fano varieties. This allows us to study these maps by studying the geometry of curves in these Fano varieties.