Noether-Lefschetz Locus and a Special Case of the Variational Hodge Conjecture: Using Elementary Techniques

Ananyo Dan ()
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Ananyo Dan: Humboldt Universität zu Berlin, Institut für Mathematik

A chapter in Analytic and Algebraic Geometry, 2017, pp 107-115 from Springer

Abstract: 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.

Keywords: Variational Hodge conjecture; Noether-Lefschetz locus; Hodge theory; non-reduced scheme; Néron-Severi group; Hilbert scheme (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-10-5648-2_7

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DOI: 10.1007/978-981-10-5648-2_7

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