The Noether–Lefschetz theorem for the divisor class group

Ravindra, G. V.; Srinivas, V.

doi:10.1016/j.jalgebra.2008.09.003

Cited by 31 publications

(26 citation statements)

References 8 publications

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“…When E is a line bundle, the above result yields the Formal Noether-Lefschetz theorem proved in [17]. Using our formalism, and a remark by M.V.Nori, we are also able to prove the following version of the Noether-Lefschetz theorem for divisor class groups (compare with the result of [17]) which generalises the main result of a paper of Joshi [12].…”

Section: Introductionsupporting

confidence: 76%

Extensions of Vector Bundles With Application to Noether–lefschetz Theorems

Ravindra

Tripathi

2013

Commun. Contemp. Math.

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Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y, we study the question of when a bundle E on X, extends to a bundle [Formula: see text] on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck–Lefschetz theory. As a consequence, we prove a Noether–Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether–Lefschetz theorems of Joshi and Ravindra–Srinivas.

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Section: Introductionsupporting

confidence: 76%

Extensions of Vector Bundles With Application to Noether–lefschetz Theorems

Ravindra

Tripathi

2013

Commun. Contemp. Math.

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“…For a generic hypersurface S described above, pullbacks of the hyperplane classes in P 1 and P 2 generate Néron-Severi lattice of S, and Picard number ρ(S) = 2 [31,32]. Let D 1 and D 2 denote divisors of hypersurface S corresponding to pullbacks of hyperplanes in P 1 and P 2 respectively.…”

Section: Class Of Elliptic K3 Surfaces Which Do Not Admit Sectionmentioning

confidence: 99%

Gauge groups and matter fields on some models of F-theory without section

Kimura

2016

J. High Energ. Phys.

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We investigate F-theory on an elliptic Calabi-Yau 4-fold without a section to the fibration. To construct an elliptic Calabi-Yau 4-fold without a section, we introduce families of elliptic K3 surfaces which do not admit a section. A product K3 × K3, with one of the K3's chosen from these families of elliptic K3 surfaces without a section, realises an elliptic Calabi-Yau 4-fold without a section. We then compactify F-theory on such K3 × K3's.We determine the gauge groups and matter fields which arise on 7-branes for these models of F-theory compactifications without a section. Since each K3 × K3 constructed does not have a section, gauge groups arising on 7-branes for F-theory models on constructed K3 × K3's do not have U(1)-part. Interestingly, exceptional gauge group E 6 appears for some cases.

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“…Actually when we wrote [4] we were unaware of [25] where a general result is proved using different techniques. Theorem 4.3.…”

Section: Noether-lefschetz Locimentioning

confidence: 99%

The Noether–Lefschetz locus of surfaces in toric threefolds

Bruzzo

Grassi

2018

Commun. Contemp. Math.

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The Noether-Lefschetz theorem asserts that any curve in a very general surface X in P 3 of degree d ≥ 4 is a restriction of a surface in the ambient space, that is, the Picard number of X is 1. We proved previously that under some conditions, which replace the condition d ≥ 4, a very general surface in a simplicial toric threefold P Σ (with orbifold singularities) has the same Picard number as P Σ . Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in P Σ in a linear system of a Cartier ample divisor with respect to a (-1)-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether-Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense.

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The Noether–Lefschetz theorem for the divisor class group

Cited by 31 publications

References 8 publications

Extensions of Vector Bundles With Application to Noether–lefschetz Theorems

Extensions of Vector Bundles With Application to Noether–lefschetz Theorems

Gauge groups and matter fields on some models of F-theory without section

The Noether–Lefschetz locus of surfaces in toric threefolds

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