We continue our study of the Noether–Lefschetz loci in toric varieties and investigate deformation of pairs (V, X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a simplicial projective toric variety PΣ2k+1 , with V⊂ X . The hypersurface X is supposed to be of Macaulay type, which means that its toric Jacobian ideal is Cox–Gorenstein, a property that generalizes the notion of Gorenstein ideal in the standard polynomial ring. Under some assumptions, we prove that the class λV∈ Hk,k(X) deforms to an algebraic class if and only if it remains of type (k, k). Actually we prove that locally the Noether–Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 in our previous work (Bruzzo and Montoya 15(2):682–694, 2021) and the main theorem proved by Dan (in: Analytic and Algebraic Geometry. Hindustan Book Agency, New Delhi, pp 107–115, 2017).

Deformation of pairs and Noether–Lefschetz loci in toric varieties / Bruzzo, U.; Montoya, W. D.. - In: EUROPEAN JOURNAL OF MATHEMATICS. - ISSN 2199-675X. - 9:4(2023), pp. 1-10. [10.1007/s40879-023-00702-4]

Deformation of pairs and Noether–Lefschetz loci in toric varieties

Bruzzo U.;Montoya W. D.
2023-01-01

Abstract

We continue our study of the Noether–Lefschetz loci in toric varieties and investigate deformation of pairs (V, X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a simplicial projective toric variety PΣ2k+1 , with V⊂ X . The hypersurface X is supposed to be of Macaulay type, which means that its toric Jacobian ideal is Cox–Gorenstein, a property that generalizes the notion of Gorenstein ideal in the standard polynomial ring. Under some assumptions, we prove that the class λV∈ Hk,k(X) deforms to an algebraic class if and only if it remains of type (k, k). Actually we prove that locally the Noether–Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 in our previous work (Bruzzo and Montoya 15(2):682–694, 2021) and the main theorem proved by Dan (in: Analytic and Algebraic Geometry. Hindustan Book Agency, New Delhi, pp 107–115, 2017).
2023
9
4
1
10
108
https://doi.org/10.1007/s40879-023-00702-4
https://arxiv.org/abs/2203.00664
Bruzzo, U.; Montoya, W. D.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/136034
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