Twistor spaces with a pencil of fundamental divisors
In this paper simply connected twistor spaces containing a pencil of fundamental divisors are studied. The Riemannian base for such spaces is diffeomorphic to the connected sum . We obtain for a complete description of the set of curves intersecting the fundamental line bundle negatively. For this purpose we introduce a combinatorial structure, called emphblow-up graph. We show that for generic the algebraic dimension can be computed by the formula . A detailed study of the anti Kodaira dimension 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 is a LeBrun space citeLeB2. We characterise such spaces also by the property to contain a smooth rational non-real curve with .
Cite this article
B. Kreußler, Twistor spaces with a pencil of fundamental divisors. Doc. Math. 4 (1999), pp. 127–166DOI 10.4171/DM/56