The Noether-Lefschetz theorem for the divisor class group

G. V. Ravindra; V. Srinivas

doi:10.1016/j.jalgebra.2008.09.003

The Noether-Lefschetz theorem for the divisor class group

G. V. Ravindra
, V. Srinivas

Mathematics

Indian Institute of Science Bangalore

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

Abstract

Let X be a normal projective threefold over a field of characteristic zero and | L | be a base-point free, ample linear system on X. Under suitable hypotheses on (X, | L |), we prove that for a very general member Y ∈ | L |, the restriction map on divisor class groups Cl (X) → Cl (Y) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X ⊂ P_C³ of degree ≥4 has Pic (X) ≅ Z.

Original language	English
Pages (from-to)	3373-3391
Number of pages	19
Journal	Journal of Algebra
Volume	322
Issue number	9
DOIs	https://doi.org/10.1016/j.jalgebra.2008.09.003
State	Published - Nov 1 2009

Keywords

Divisor class group
Noether-Lefschetz theorem

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10.1016/j.jalgebra.2008.09.003

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abstract = "Let X be a normal projective threefold over a field of characteristic zero and | L | be a base-point free, ample linear system on X. Under suitable hypotheses on (X, | L |), we prove that for a very general member Y ∈ | L |, the restriction map on divisor class groups Cl (X) → Cl (Y) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X ⊂ PC3 of degree ≥4 has Pic (X) ≅ Z.",

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N2 - Let X be a normal projective threefold over a field of characteristic zero and | L | be a base-point free, ample linear system on X. Under suitable hypotheses on (X, | L |), we prove that for a very general member Y ∈ | L |, the restriction map on divisor class groups Cl (X) → Cl (Y) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X ⊂ PC3 of degree ≥4 has Pic (X) ≅ Z.

AB - Let X be a normal projective threefold over a field of characteristic zero and | L | be a base-point free, ample linear system on X. Under suitable hypotheses on (X, | L |), we prove that for a very general member Y ∈ | L |, the restriction map on divisor class groups Cl (X) → Cl (Y) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X ⊂ PC3 of degree ≥4 has Pic (X) ≅ Z.

KW - Divisor class group

KW - Noether-Lefschetz theorem

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JO - Journal of Algebra

JF - Journal of Algebra

IS - 9

ER -

The Noether-Lefschetz theorem for the divisor class group

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