Twistor spaces with a pencil of fundamental divisors

Abstract

In this paper simply connected twistor spaces $Z$ containing a pencil of fundamental divisors are studied. The Riemannian base for such spaces is diffeomorphic to the connected sum $n{\mathbb{CP}}^2$. We obtain for $n\ge 5$ a complete description of the set of curves intersecting the fundamental line bundle $K^{-}\frac{1}{2}$ negatively. For this purpose we introduce a combinatorial structure, called blow-up graph. We show that for generic $S\in \mid -\frac{1}{2}K\mid$ the algebraic dimension can be computed by the formula $a(Z)=1+{\kappa }^{-1}(S)$. A detailed study of the anti Kodaira dimension ${\kappa }^{-1}(S)$ of rational surfaces permits to read off the algebraic dimension from the blow-up graphs. This gives a characterisation of Moishezon twistor spaces by the structure of the corresponding blow-up graphs. We study the behaviour of these graphs under small deformations. The results are applied to prove the main existence result, which states that every blow-up graph belongs to a fundamental divisor of a twistor space. We show, furthermore, that a twistor space with $\dim \mid -\frac{1}{2}K\mid =3$ is a LeBrun space [LeB2]. We characterise such spaces also by the property to contain a smooth rational non-real curve $C$ with $C.(-\frac{1}{2}K)=2-n$.