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

TL;DR

This paper investigates the degree of irrationality of smooth surfaces in projective 3-space, establishing conditions under which it remains invariant or decreases when considering products or dominating varieties.

Contribution

It proves that the degree of irrationality of such surfaces remains unchanged under product with projective spaces and characterizes when it decreases for dominating varieties.

Findings

irr(S×P^m)=irr(S) for all m

irr(Y)<irr(S) iff S contains a rational curve

Provides criteria for irrationality behavior in algebraic surfaces

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.