Algebraic dimension of twistor spaces whose fundamental system is a pencil

TL;DR

This paper proves that for twistor spaces over n#CP^2 with n>4, the algebraic dimension cannot be two if the fundamental system is a pencil, implying such spaces have limited algebraic complexity.

Contribution

It establishes a restriction on the algebraic dimension of twistor spaces with a pencil as the fundamental system for n>4.

Findings

Algebraic dimension cannot be two for n>4 when fundamental system is a pencil.

If algebraic dimension is two, the fundamental system is either empty or a single member.

Open problem remains for existence of such twistor spaces with algebraic dimension two.

Abstract

We show that the algebraic dimension of a twistor space over n#CP^2 cannot be two if n>4 and the fundamental system (i.e. the linear system associated to the half-anti-canonical bundle, which is available on any twistor space) is a pencil. This means that if the algebraic dimension of a twistor space on n#CP^2, n>4, is two, then the fundamental system either is empty or consists of a single member. The existence problem for a twistor space on n#CP^2 with algebraic dimension two is open for n>4.