For a quasi-smooth hypersurface X in a projective simplicial toric variety PΣ, the morphism i∗: Hp(PΣ) → Hp(X) induced by the inclusion is injective for p= dim X and an isomorphism for p< dim X- 1. This allows one to define the Noether–Lefschetz locus NL β as the locus of quasi-smooth hypersurfaces of degree β such that i∗ acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if dim PΣ= 2 k+ 1 and kβ- β= nη, n∈ N, where η is the class of a 0-regular ample divisor, and β is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree β satisfies the bounds n+1⩽codimZ⩽hk-1,k+1(X).

Codimension bounds for the Noether–Lefschetz components for toric varieties / Bruzzo, U.; Montoya, W. D.. - In: EUROPEAN JOURNAL OF MATHEMATICS. - ISSN 2199-6768. - (In corso di stampa), pp. 1-9. [10.1007/s40879-021-00461-0]

Codimension bounds for the Noether–Lefschetz components for toric varieties

Bruzzo, U.
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In corso di stampa

Abstract

For a quasi-smooth hypersurface X in a projective simplicial toric variety PΣ, the morphism i∗: Hp(PΣ) → Hp(X) induced by the inclusion is injective for p= dim X and an isomorphism for p< dim X- 1. This allows one to define the Noether–Lefschetz locus NL β as the locus of quasi-smooth hypersurfaces of degree β such that i∗ acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if dim PΣ= 2 k+ 1 and kβ- β= nη, n∈ N, where η is the class of a 0-regular ample divisor, and β is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree β satisfies the bounds n+1⩽codimZ⩽hk-1,k+1(X).
1
9
10.1007/s40879-021-00461-0
https://arxiv.org/abs/2001.01960
Bruzzo, U.; Montoya, W. D.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/125833
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