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Twistor spaces and Fano threefolds. (English) Zbl 0817.14019
Usually, twistor spaces are certain complex 3–manifolds fibred over a Riemannian 4-manifold. It was shown by N. J. Hitchin [Proc. Lond. Math. Soc., III. Ser. 43, 133-150 (1981; Zbl 0474.14024)] and by Th. Friedrich and H. Kurke [Math. Nachr. 106, 271-299 (1982; Zbl 0503.53035)] that only two of these spaces are projective algebraic. They are Fano varieties. The author of the paper under review relaxes the usual definition of twistor space as follows: he calls “twistor space” every complex projective algebraic 3-fold containing a twistor curve, that is a smooth rational curve with normal bundle $${\mathcal O}(1)\oplus{\mathcal O}(1)$$. If there exists an irreducible family of such twistor curves sweeping out the whole twistor space, then he calls it “minimal”. The two usual projective twistor spaces are, of course, minimal twistor spaces in the sense of the paper. These more general twistor spaces have many properties of usual twistor spaces. The author studies the following question:
Is a twistor space a Fano variety? Which Fano varieties arise in this way? Let $$X$$ be a twistor space (in the sense of this paper) and $$X \subset \mathbb{P}_ n$$ an embedding such that the twistor curves have degree $$d$$. He shows that $$n < {d + 3 \choose 3}$$. In particular, he obtains: the only twistor space admitting a projective embedding with $$d = 1$$ is $$\mathbb{P}_ 3$$. Then he focuses on the case $$d = 2$$, the case of “twistor conics”. He obtains the following theorem: if $$X \subset \mathbb{P}_ n$$ is a twistor space in which twistor curves are conics (case $$d = 2)$$, then one of the following holds:
(i) $$n = 9$$, $$X \cong \mathbb{P}_ 3$$ and $$X \subset \mathbb{P}_ 9$$ is the Veronese embedding;
(ii) $$n \leq 8$$, $$X$$ is a Fano 3-fold of index 2;
(iii) $$n = 6$$ and $$X$$ is a quartic 3-fold whose generic hyperplane section is a rational scroll.
The author achieves this result by studying the generic hyperplane section of $$X$$. An example of case (iii) is obtained by the birational image of $$\mathbb{P}_ 3$$ under the rational map $$\mathbb{P}_ 3 \to \mathbb{P}_ 6$$ given by the linear system of quadrics through a line in $$\mathbb{P}_ 3$$. This defines a twistor space which is not minimal. The author conjectures that there is no other threefold in case (iii). In the rest of the paper he studies case (ii). The result is a complete list of index 2 Fano threefolds which are twistor spaces.

##### MSC:
 14J30 $$3$$-folds 32L25 Twistor theory, double fibrations (complex-analytic aspects) 14J45 Fano varieties
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