Twistor spaces with a pencil of fundamental divisors

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^-\frac12 negatively. For this purpose we introduce a combinatorial structure, called blow-up graph . We show that for generic S\in\mid-\frac12 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-\frac12 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.(-\frac12 K)=2-n .