On irrationality of surfaces in $\mathbb{P}^3$

Abstract

The degree of irrationality $irr(X)$ of a $n$-dimensional complex projective variety $X$ is the least degree of a dominant rational map $X\dashrightarrow \mathbb{P}^n$. It is a well-known fact that given a product $X\times \mathbb{P}^m$ or a $n$-dimensional variety $Y$ dominating $X$, their degrees of irrationality may be smaller than the degree of irrationality of $X$. In this paper, we focus on smooth surfaces $S\subset\mathbb{P}^3$ of degree $d\geq 5$, and we prove that $irr(S\times\mathbb{P}^{m})=irr(S)$ for any positive integer $m$, whereas $irr(Y)<irr(S)$ occurs for some $Y$ dominating $S$ if and only if $S$ contains a rational curve.