Codimension bounds for the Noether–Lefschetz components for toric varieties

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).