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 $X⇢{\mathbb{P}}^n$. 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 $d\ge 5$, 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.

Final version. 10 pages

References in corpus (2)

• Measures of irrationality for hypersurfaces of large degree
• Gonality of curves on fundamental loci of first order congruences (Appendix to article of Ein, Lazarsfeld and Ullery)

Cited by in corpus (4)

• Degree of irrationality of very general abelian surfaces
• Gonality of curves on general hypersurfaces
• Tilt stability and the degree of irrationality of surfaces on threefolds
• On irrationality of hypersurfaces In ${\mathbf{P}}^{n+1}$