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

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 $d\ge 2n+2$, 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 $m\ge 1$.

7 pages. Comments are welcome! In the latest version, Schreieder's result on stable rationality of hypersurfaces is added