# Twistor spaces with a pencil of fundamental divisors

### B. Kreußler

## 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 $nCP_{2}$. We obtain for $n≥5$ a complete description of the set of curves intersecting the fundamental line bundle $K_{−}21 $ negatively. For this purpose we introduce a combinatorial structure, called *blow-up graph*. We show that for generic $S∈∣−21 K∣$ the algebraic dimension can be computed by the formula $a(Z)=1+κ_{−1}(S)$. A detailed study of the anti Kodaira dimension $κ_{−1}(S)$ 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 $dim∣−21 K∣=3$ is a LeBrun space [LeB2]. We characterise such spaces also by the property to contain a smooth rational non-real curve $C$ with $C.(−21 K)=2−n$.

## Cite this article

B. Kreußler, Twistor spaces with a pencil of fundamental divisors. Doc. Math. 4 (1999), pp. 127–166

DOI 10.4171/DM/56