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

In: Analytic and Algebraic Geometry

Author

Listed:
  • Ananyo Dan

    (Humboldt Universität zu Berlin, Institut für Mathematik)

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.

Suggested Citation

  • Ananyo Dan, 2017. "Noether-Lefschetz Locus and a Special Case of the Variational Hodge Conjecture: Using Elementary Techniques," Springer Books, in: Anilatmaja Aryasomayajula & Indranil Biswas & Archana S. Morye & A. J. Parameswaran (ed.), Analytic and Algebraic Geometry, pages 107-115, Springer.
    • Handle: RePEc:spr:sprchp:978-981-10-5648-2_7
    DOI: 10.1007/978-981-10-5648-2_7
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