On irrationality of hypersurfaces In $\mathbf{P}^{n+1}$

TL;DR

This paper investigates measures of irrationality for hypersurfaces in projective space, confirming a conjecture for very general smooth hypersurfaces of certain degrees and establishing invariance of irrationality under product with projective space.

Contribution

It proves that for very general smooth hypersurfaces with degree at least 2n+2, the stable and universal irrationality measures equal d-1, and shows irrationality remains unchanged when taking products with projective spaces.

Findings

Confirmed conjecture on irrationality measures for hypersurfaces of degree ≥ 2n+2.

Proved that irrationality is invariant under product with projective space.

Established equality of different irrationality measures for the specified hypersurfaces.

Abstract

In this note, we study various measures of irrationality for hypersurfaces in projective spaces which were recently proposed by Bastianelli, De Poi, Ein, Lazarsfeld and Ullery. In particular, we answer the question raised by Bastianelli that if $X \subset P^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $d\geq 2n+2$, then $\text{stab.irr}(X)=\text{uni.irr}(X)=d-1$. As a corollary, we prove that $\text{irr}(X\times P^{m})=\text{irr}(X)$ for any integer $m\geq 1$.