記述言語：英語   出版者・発行元：WALTER DE GRUYTER & CO  

In recent papers [10], [11], we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called minitwistor spaces). These include the twistor spaces studied in the papers as very special cases.
Our source of the result is a series of self-dual metrics with torus action constructed by D. Joyce [15]. Actually, for arbitrary Joyce metrics and U(1)-subgroups of the torus which fixes a torus-invariant 2-sphere, we first determine the associated minitwistor spaces in explicit forms. Next by analyzing the meromorphic maps from the twistor spaces to the minitwistor spaces, we realize projective models of the twistor spaces of all Joyce metrics, as conic bundles over the minitwistor spaces. Then we prove that for any one of these minitwistor spaces, there exist Moishezon twistor spaces with only C*-action whose quotient space is the given minitwistor space. This result generates numerous Moishezon twistor spaces which cannot be found in the literature (including the author's papers), in quite explicit form.

DOI： 10.1515/CRELLE.2010.041

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記述言語：英語   出版者・発行元：MATH SOC JAPAN  

In this note, we explicitly construct the twister spaces of some Joyce metrics oil the connected sum of arbitrary number of complex projective planes. Unlike our former construction for the case of four complex projective planes, the present construction mainly utilizes minitwistor spaces, and partially follows the method and construction given in [5] and [6].

DOI： 10.2969/jmsj/06141243

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記述言語：英語   出版者・発行元：INT PRESS  

In this paper we explicitly construct Moishezon twistor spaces on nCP(2) for arbitrary n >= 2 which admit a holomorphic C*-action. When n = 2, they coincide with Y. Poon's twistor spaces. When n = 3, they coincide with the ones studied by the author in [14]. When n >= 4, they are new twistor spaces, to the best of the author's knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C*-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP(4). It turns out that the structure of the minitwistor space is independent of n. Further, we explicitly construct a CP(2)-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP(1) x CP(1) in LeBrun's case is replaced by our minitwistor spaces found in [15].

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記述言語：英語   出版者・発行元：SPRINGER  

In a recent paper ([9]) we constructed a series of new Moishezon twistor spaces which are a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on n CP 2 for arbitrary n 3, which can be regarded as a generalization of the twistor spaces of 'double solid type' on 3CP 2 studied by Kreuuler, Kurke, Poon and the author. Similarly to the twistor spaces of 'double solid type' on 3CP2 , projective models of the present twistor spaces have a natural structure of double covering of a CP2 . We explicitly give a defining polynomial of the branch divisor of the double covering, whose restriction to fibers is degree four. If n >= 4 these are new twistor spaces, to the best of the author's knowledge. We also compute the dimension of the moduli space of these twistor spaces. Differently from [9], the present investigation is based on analysis of pluri-(half-)anticanonical systems of the twistor spaces.

DOI： 10.1007/s00222-008-0139-5

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記述言語：英語   出版者・発行元：UNIV PRESS INC  

We explicitly construct the twistor spaces of some self-dual metrics with torus action given by ID. Joyce. Starting from a fiber space over a projective line whose fibers are compact singular toric surfaces, we apply a number of birational transformations to obtain the desired twistor spaces. These constructions are based on a detailed analysis of the anticanonical system of the twistor spaces.

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記述言語：英語   出版者・発行元：INT PRESS  

We determine a global structure of the moduli space of self-dual metrics on 3CP(2) satisfying the following three properties: (i) the scalar curvature is of positive type, (ii) they admit a non-trivial Killing field, (iii) they are not conformal to the LeBrun's self-dual metrics based on the 'hyperbolic ansatz'. We prove that the moduli space of these metrics is isomorphic to an orbifold R-3/G, where G is an involution of R-3 having two-dimensional fixed locus. In particular, the moduli space is non-empty and connected. We also remark that Joyce's self-dual metrics with torus symmetry appear as a limit of our self-dual metrics.
Our proof of the result is based on the twistor theory. We first determine a defining equation of a projective model of the twistor space of the metric, and then prove that the projective model is always birational to a twistor space, by determining the family of twistor lines. In determining them, a key role is played by a classical result in algebraic geometry that a smooth plane quartic always possesses twenty-eight bitangents.

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記述言語：英語   出版者・発行元：KINOKUNIYA CO LTD  

We give an explicit description of rational curves in the product of three copies of complex projective lines, which are transformed into twistor lines in M. Nagata's example of non-projective complete algebraic variety, viewed as the twistor space of Eguchi-Hanson metric. In particular, we show that there exist two families of such curves and both of them are parameterized by mutually diffeomorphic, connected real 4-dimensional manifolds. We also give a relationship between these two families through a birational transformation naturally associated to the Nagata's example.

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DOI： 10.1090/S0002-9939-06-08489-9

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記述言語：英語   出版者・発行元：AMER MATHEMATICAL SOC  

We show that a twistor space of a self-dual metric on 4CP(2) with U(1)-isometry is not Moishezon iff there is a C*-orbit biholomorphic to a smooth elliptic curve, where the C*-action is the complexification of the U( 1)action on the twistor space. It follows that the U(1)-isometry has a two-sphere whose isotropy group is Z(2). We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreu ss ler is affirmative even though a twistor space is required to have a non-trivial automorphism group.

DOI： 10.1090/S0002-9947-05-04141-3

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