Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids

Hadan, Ingo

doi:10.1002/(sici)1522-2616(200002)210:1<127::aid-mana127>3.0.co;2-c

Cited by 6 publications

(19 citation statements)

References 20 publications

(10 reference statements)

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“…Let Φ : Z → CP 3 and B ⊂ CP 3 be as above. Then there exists a homogeneous coordinate (y 0 : y 1 : y 2 : y 3 ) on CP 3 fulfilling (i)-(iii) below: (i) a defining equation of B is given by (3) (y 2 y 3 + Q(y 0 , y 1 )) 2 − y 0 y 1 (y 0 + y 1 )(y 0 − ay 1 ) = 0,…”

Section: Defining Equations Of the Branch Quartic Surfaces And Their mentioning

confidence: 99%

“…The following result will be also needed in the next section: Proposition 2.9. Let B be a quartic surface defined by (3). Then B does not contain real lines.…”

Section: Defining Equations Of the Branch Quartic Surfaces And Their mentioning

confidence: 99%

“…As H λ is defined by x 0 = λx 1 , Φ −1 0 (H λ ) is locally defined by ξη = f (λ)x 4 1 around p ∞ . Substituting ξ = uℓ 1 for the first blow-up, the inverse image becomes uη = (f (λ)/ℓ 1 (λ, 1))x 3 1 . Next substituting u = vℓ 2 for the second blow-up, we get vη = (f (λ)/ℓ 1 (λ, 1)ℓ 2 (λ, 1))x 2 1 .…”

Section: So We Havementioning

confidence: 99%

“…Now we explain why the involution G appears in Theorem 1.1. In the course of our proof of Theorem 1.1, we construct a family M of self-dual metrics on 3CP 2 parameterized by R 3 , which contains all metrics satisfying conditions (i)-(iii). But it will turned out that the family does not effectively parameterize the metrics; nemaly we see that an involution G acts on this parameter space R 3 in such a way that two points exchanged by the involution represent the same conformal class.…”

Section: Introductionmentioning

confidence: 99%

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Self-dual metrics and twenty-eight bitangents

Honda¹

2007

J. Differential Geom.

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We study self-dual metrics on 3CP 2 of positive scalar curvature admitting a non-zero Killing field, but which are not conformally isometric to LeBrun's metrics. Firstly, we determine defining equations of the twistor spaces of such self-dual metrics. Next we prove that conversely, the complex threefolds defined by the equations always become twistor spaces of self-dual metrics on 3CP 2 of the above kind. As a corollary, we determine a global structure of the moduli spaces of these self-dual metrics; namely we show that the moduli space is naturally identified with an orbifold R 3 /G, where G is an involution of R 3 having two-dimensional fixed locus. Combined with works of LeBrun, this settles a moduli problem of self-dual metrics on 3CP 2 of positive scalar curvature admitting a non-trivial Killing field. In particular, it is shown that any two self-dual metrics on 3CP 3 of positive scalar curvature admitting a non-zero Killing field can be connected by deformation keeping the self-duality. In our proof, 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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Section: Defining Equations Of the Branch Quartic Surfaces And Their mentioning

confidence: 99%

“…The following result will be also needed in the next section: Proposition 2.9. Let B be a quartic surface defined by (3). Then B does not contain real lines.…”

Section: Defining Equations Of the Branch Quartic Surfaces And Their mentioning

confidence: 99%

Section: So We Havementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Self-dual metrics and twenty-eight bitangents

Honda¹

2007

J. Differential Geom.

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“…Hadan [13] proves that -to any smooth plane quartic curve, there are 63 disjoint one parameter families of smooth plane conics (simple) tangentially meeting the quartic at 4 points. Using this, we obtain that a general conic in each such family gives rise to a pair of Ulrich line bundles.…”

Section: Introductionmentioning

confidence: 99%

Ulrich line bundles on double planes

Parameswaran,

Narayanan

2019

Preprint

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Consider a smooth complex surface X which is a double cover of the projective plane P 2 branched along a smooth curve of degree 2s. In this article, we study the geometric conditions which are equivalent to the existence of Ulrich line bundles on X with respect to this double covering. Also, for every s ≥ 1, we describe the classes of such surfaces which admit Ulrich line bundles and give examples.

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On integral points of some Fano threefolds and their Hilbert schemes of lines and conics

Corvaja

Zucconi

2023

Rend. Circ. Mat. Palermo, II. Ser

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We prove some density results for integral points on affine open sets of Fano threefolds. For instance, let $$X^o=\mathbb P^3{\setminus } D$$ X o = P 3 \ D where D is the union of two quadrics such that their intersection contains a smooth conic, or the union of a smooth quadric surface and two planes, or the union of a smooth cubic surface V and a plane $$\Pi $$ Π such that the intersection $$V\cap \Pi $$ V ∩ Π contains a line. In all these cases we show that the set of integral points of $$X^o$$ X o is potentially dense. We apply the above results to prove that integral points are potentially dense in some log-Fano or in some log-Calabi-Yau threefold.

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Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids

Cited by 6 publications

References 20 publications

Self-dual metrics and twenty-eight bitangents

Self-dual metrics and twenty-eight bitangents

Ulrich line bundles on double planes

On integral points of some Fano threefolds and their Hilbert schemes of lines and conics

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