On fields of meromorphic functions on neighborhoods of 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 $\mathbb{C}$.

15 pages, 1 figure; version 2: bibliography extended, appropiate changes in the text; version 3: minor improvements