Codimension bounds for the Noether-Lefschetz components for toric varieties (Q2087687)

Fix a projective simplicial toric variety \(\mathbb{P}_{\Sigma}\) of odd dimension and an ample class \(\beta\in \mathrm{Pic}(\mathbb{P}_\Sigma)\). This article studies the Noether-Lefschetz locus for degree \(\beta\) hypersurfaces in \(\mathbb{P}_{\Sigma}\). (Classically the Noether-Lefschetz locus is the subscheme of the hypersurface parameter space where the Picard number is greater than the Picard number of the ambient variety. The definition of the Noether-Lefschetz locus was extended by the authors to quasi-smooth hypersurfaces in a projective simplicial toric variety in [\textit{U. Bruzzo} and \textit{W. D. Montoya}, ``An asymptotic description of the Noether-Lefschetz components in toric varieties'', Preprint, \url{arXiv:1905.01570v3}]). The main result of this article is a bound of the codimension for the irreducible components of the Noether-Lefschetz locus. The lower bound is proved by the ``explicit Noether-Lefschetz theorem for toric varieties.'' The upper bound is proved by the Hodge theory.