# Fano hypersurfaces with arbitrarily large degrees of irrationality

## TL;DR

This paper demonstrates that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality, providing new examples of rationally connected varieties with high irrationality degrees.

## Contribution

It establishes the first known examples of rationally connected varieties with irrationality degrees exceeding 3, using degeneration techniques in characteristic p.

## Key findings

- Degree of irrationality grows at least as a constant times sqrt(n)
- Introduces a method to analyze how irrationality behaves in families of varieties
- Shows that certain invariants only decrease on special fibers

## Abstract

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index e, then the degree of irrationality of a very general complex Fano hypersurface of index e and dimension n is bounded from below by a constant times $\sqrt{n}$. To our knowledge this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic p argument which Koll\'ar used to prove nonrationality of Fano hypersurfaces. Along the way we show that in a family of varieties, the invariant "the minimal degree of a dominant rational map to a ruled variety" can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties the degree of irrationality also behaves well under specialization.