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 nCP2 . We obtain for n 5 a complete description of the set of curves intersecting the fundamental line bundle K-1/2 negatively. For this purpose we introduce a combinatorial structure, called blow-up graph. We show that for generic S 2j-1/2K j the algebraic dimension can be computed by the formula a(Z) = 1 + K-1(S). A detailed study of the anti Kodaira dimension K-1(S) of rational surfaces permits to read o 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 -1/2K = 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:(-1/2K) = 2-n.