On the algebraic dimension of twistor spaces over the connected sum of four complex projective planes

TL;DR

This paper investigates the algebraic dimension of twistor spaces over four complex projective planes, establishing conditions for when such spaces are Moishezon and analyzing their linear systems and geometric properties.

Contribution

It provides a characterization of Moishezon twistor spaces over 4P^2 based on the nefness of the anticanonical class and describes the structure of their fundamental linear system.

Findings

Twistor space is Moishezon iff its anticanonical class is not nef.

Dimension of |−1/2K| is at most the algebraic dimension of the space.

Spaces with a pencil of divisors of degree one have |−1/2K| dimension equal to 3.

Abstract

We study the algebraic dimension of twistor spaces of positive type over $4\bbfP^2$. We show that such a twistor space is Moishezon if and only if its anticanonical class is not nef. More precisely, we show the equivalence of being Moishezon with the existence of a smooth rational curve having negative intersection number with the anticanonical class. Furthermore, we give precise information on the dimension and base locus of the fundamental linear system $|{-1/2}K|$. This implies, for example, $\dim|{-1/2}K|\leq a(Z)$. We characterize those twistor spaces over $4\bbfP^2$, which contain a pencil of divisors of degree one by the property $\dim|{-1/2}K| = 3$.