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

Abstract:

The purpose of this note is to study various measures of irrationality for hypersurfaces in projective spaces which were proposed recently by F. Bastianelli et al. In particular, we answer the question raised by Bastianelli that if $X\subset \mathbf {P}^{n+1}$ 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 $\text {irr}(X\times \mathbf {P}^m)=\text {irr}(X)$ for any integer $m\geq 1$.