Algebraic Surfaces With Infinitely Many Twistor Lines

Altavilla, Amedeo; Ballico, Edoardo

doi:10.1017/s0004972719000534

Cited by 5 publications

(5 citation statements)

References 17 publications

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“…There are close analogues of our results with those of [35] on the Penrose fibration P 3 → S 4 . This is to be expected since the twistor spaces P 3 and F incorporate an open orbit of a complex Heisenberg group, and are birationally equivalent, a fact that extends to any two Wolf spaces of the same dimension [13].…”

Section: Introductionsupporting

confidence: 84%

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Twistor geometry of the Flag manifold

Altavilla¹,

Ballico²,

Brambilla³

et al. 2021

Preprint

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A study is made of algebraic curves and surfaces in the flag manifold F = SU (3)/T 2 , and their configuration relative to the twistor projection π from F to the complex projective plane P 2 , defined with the help of an anti-holomorphic involution j. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space P 3 of S 4 . Deformations of twistor fibres project to real surfaces in P 2 , whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1, 1). These surfaces define orthogonal complex structures on specified dense open subsets of P 2 relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree (1, 1) are determined, and bounds given on the number of twistor fibres that are contained in more general algebraic surfaces in F.

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Section: Introductionsupporting

confidence: 84%

“…The involution j maps J to −J and, in the twistor context, j-invariant objects are called 'real'. In the analogous situation of the Penrose fibration P 3 → S 4 , there has been extensive study of algebraic surfaces in P 3 and their associated orthogonal complex structures in domains of S 4 [35,2,3,4,5,15,6,7,21,8].…”

Section: Introductionmentioning

confidence: 99%

Twistor geometry of the Flag manifold

Altavilla¹,

Ballico²,

Brambilla³

et al. 2021

Preprint

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“…These results resemble the behavior of the other basic example of algebraic twistor space, which is P 3 projecting onto S 4 . In that case, considering the results in [11,2], we obtained that the only smooth case of surface containing infinitely many twistor fibers is that of quadrics, while the other admissible cases are only ruled surfaces of even degree. Here, in a sense, the analogy is complete, as the "total degree" in the Segre embedding of a surface of bidegree (a, b) is a + b.…”

Section: Introductionmentioning

confidence: 93%

“…In particular κ(S) = −∞ if S contains infinitely many twistor fibers. Notice that the same argument can be adapted to the case of surfaces in P 3 , seen as twistor space of S 4 , containing infinitely many twistor fibers [2].…”

Section: Surfaces With Infinitely Many Twistor Fibersmentioning

confidence: 99%

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Surfaces in the flag threefold containing smooth conics and twistor fibers

Altavilla,

Ballico,

Brambilla

2022

Preprint

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We study smooth integral curves of bidegree (1, 1), called smooth conics, in the flag threefold F. The study is motivated by the fact that the family of smooth conics contains the set of fibers of the twistor projection F → CP 2 . We give a bound on the maximum number of smooth conics contained in a smooth surface S ⊂ F. Then, we show qualitative properties of algebraic surfaces containing a prescribed number of smooth conics. Lastly, we study surfaces containing infinitely many twistor fibers. We show that the only smooth cases are surfaces of bidegree (1, 1). Then, for any integer a > 1, we exhibit a method to construct an integral surface of bidegree (a, a) containing infinitely many twistor fibers.

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