Moishezon twistor spaces on 4ℂℙ²

Honda, Nobuhiro

doi:10.1090/s1056-3911-2013-00619-0

J. Algebraic Geom.

2013

DOI: 10.1090/s1056-3911-2013-00619-0

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Moishezon twistor spaces on 4ℂℙ²

Nobuhiro Honda

Abstract: In this paper we classify all Moishezon twistor spaces on 4CP 2 . The classification is given in terms of the structure of the anticanonical system of the twistor spaces. We show that the anticanonical map satisfies one of the following three properties: (a) birational over the image, (b) two to one over the image, or (c) the image is two-dimensional. We determine structure of the images for each case in explicit forms. Then we intensively investigate structure of the twistor spaces in the case (b), and determ… Show more

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2023

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Twistors, Quartics, and del Pezzo Fibrations

Honda¹

2023

Memoirs of the AMS

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It has been known that twistor spaces associated to self-dual metrics on compact 4-manifolds are source of interesting examples of non-projective Moishezon threefolds. In this paper we investigate the structure of a variety of new Moishezon twistor spaces. The anti-canonical line bundle on any twistor space admits a canonical half, and we analyze the structure of twistor spaces by using the pluri-half-anti-canonical map from the twistor spaces. Specifically, each of the present twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of rational curves, and it is an anti-canonical curve on smooth members of the pencil. These twistor spaces are naturally classified into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic hypersurface has to be of a specific form. Together with our previous result in \cite{Hon_{C}re1}, the present result completes a classification of Moishezon twistor spaces whose half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is smaller than a pencil. Twistor spaces which have a similar structure were studied in \cite{Hon_{I}nv} and \cite{Hon_{C}re2}, and they are very special examples among the present twistor spaces.

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Twistors, Quartics, and del Pezzo Fibrations

Honda¹

2023

Memoirs of the AMS

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Moishezon twistor spaces on 4ℂℙ²

Cited by 1 publication

References 24 publications

Twistors, Quartics, and del Pezzo Fibrations

Twistors, Quartics, and del Pezzo Fibrations

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