Algebraic dimension of twistor spaces whose fundamental system is a pencil (pre-published version)

Abstract

We show that the algebraic dimension of a twistor space over nℂℙ2 cannot be two if n>4 and the fundamental system (that is, 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 nℂℙ2, n>4, is two, then the fundamental system is either empty or consists of a single member. The existence problem for a twistor space on nℂℙ2 with algebraic dimension two is open for n>4.