Twistor spaces with a pencil of fundamental divisors

  • B. Kreußler

Twistor spaces with a pencil of fundamental divisors cover
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In this paper simply connected twistor spaces ZZ containing a pencil of fundamental divisors are studied. The Riemannian base for such spaces is diffeomorphic to the connected sum n\PPn\PP. We obtain for n5n\ge 5 a complete description of the set of curves intersecting the fundamental line bundle \fb1\fb{1} negatively. For this purpose we introduce a combinatorial structure, called emphblow-up graph. We show that for generic S\fundS\in\mid\fund\mid the algebraic dimension can be computed by the formula a(Z)=1+κ1(S)a(Z)=1+\kappa^{-1}(S). A detailed study of the anti Kodaira dimension κ1(S)\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\fund=3\dim\mid\fund\mid=3 is a LeBrun space citeLeB2. We characterise such spaces also by the property to contain a smooth rational non-real curve CC with C.(\fund)=2nC.(\fund)=2-n.

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

DOI 10.4171/DM/56