On fields of meromorphic functions on neighborhoods of rational curves

TL;DR

This paper investigates the structure of meromorphic function fields on complex surfaces containing rational curves, establishing conditions under which these fields are isomorphic to rational function fields in one or two variables.

Contribution

It proves that the existence of a non-constant meromorphic function on such surfaces implies their meromorphic function field is isomorphic to a rational function field in one or two variables.

Findings

If a complex surface contains a rational curve with positive self-intersection, then its meromorphic function field is rational.

The field of meromorphic functions is isomorphic to either rac{}{(x)} or rac{}{(x,y)}.

The result generalizes the understanding of meromorphic functions on non-compact complex surfaces with rational curves.

Abstract

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the field of meromorphic functions on $F$ is isomorphic to the field of rational functions in one or two variables over $\mathbb C$.