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Noether-Lefschetz Locus and a Special Case of the Variational Hodge Conjecture: Using Elementary Techniques
- Source :
- Analytic and Algebraic Geometry ISBN: 9789811056482
- Publication Year :
- 2017
- Publisher :
- Springer Singapore, 2017.
-
Abstract
- Fix integers n ≥ 1 and d such that nd > 2n + 2. The Noether-Lefschetz locus NL d,n parametrizes smooth projective hypersurfaces of degree d in ℙ2n+1 satisfying the condition: H n,n (X,ℂ) ∩ H2n(X,ℚ) ≠ ℚ. An irreducible component of the Noether-Lefschetz locus is locally a Hodge locus. One question is to ask under what choice of a Hodge class γ∈ H n,n (X,ℂ) ∩ H2n(X, ℚ) does the variational Hodge conjecture hold true? In this article we use methods coming from commutative algebra and Hodge theory to give an affirmative answer in the case γ is the class of a complete intersection subscheme in X of codimension n. Another problem studied in this article is: In the case n = 1 when is an irreducible component of the Noether-Lefschetz locus nonreduced? Using the theory of infinitesimal variation of Hodge structures of hypersurfaces in ℙ3, we determine all non-reduced components with codimension less than or equal to 3d for d ≫ 0. Here again our primary tool is commutative algebra.
Details
- ISBN :
- 978-981-10-5648-2
- ISBNs :
- 9789811056482
- Database :
- OpenAIRE
- Journal :
- Analytic and Algebraic Geometry ISBN: 9789811056482
- Accession number :
- edsair.doi...........cb4b03c9a82317492f9046397aa50824