On subfields of the function field of a general surface in ${\mathbb P}^3$

TL;DR

This paper investigates the existence of dominant rational maps from very general surfaces in projective 3-space to other surfaces, concluding such maps only occur when the target surface is rational or birational to the source.

Contribution

It establishes new restrictions on dominant rational maps from general surfaces in ${f P}^3$, using classification, Hodge, and deformation theories.

Findings

No dominant rational map exists unless the target is rational.

Surfaces with $p_g=q=0$ cannot be dominated by very general degree $ extgreater=5$ surfaces.

Results restrict possible mappings between algebraic surfaces.

Abstract

In this paper we study birational immersions from a very general smooth plane curve to a non-rational surface with $p_g=q=0$ to treat dominant rational maps from a very general surface $X$ of degree$\geq 5$ in ${\mathbb P}^3$ to smooth projective surfaces $Y$. Based on the classification theory of algebraic surfaces, Hodge theory, and deformation theory, we prove that there is no dominant rational map from $X$ to $Y$ unless $Y$ is rational or $Y$ is birational to $X$.