The Rational Hull of Rudin's Klein Bottle

TL;DR

This paper develops a method to compute the rational hulls of fibered sets in complex two-space and applies it to explicitly determine the rational hull of Rudin's Klein bottle, a nonorientable surface.

Contribution

It introduces a general approach for calculating rational hulls of fibered sets and provides the first explicit computation of the rational hull of Rudin's Klein bottle.

Findings

Rudin's Klein bottle has a two-dimensional rational hull.

The rational hull differs from the polynomial hull, which contains an open set.

The method applies to other surfaces in a72.

Abstract

In this note, a general result for determining the rational hulls of fibered sets in $\mathbb{C}^2$ is established. We use this to compute the rational hull of Rudin's Klein bottle, the first explicit example of a totally real nonorientable surface in $\mathbb{C}^2$. In contrast to its polynomial hull, which was shown to contain an open set by the first author in 2012, its rational hull is shown to be two-dimensional. Using the same method, we also compute the rational hulls of some other surfaces in $\mathbb{C}^2$.