Codimension bounds for the Noether-Lefschetz components for toric varieties

For a quasi-smooth hyper-surface $X$ in a projective simplicial toric variety $P$, the morphism $i:H^p(P) \to H^p(X)$ induced by the inclusion is injective for $p=d$ and an isomorphism for $p<d-1$, where $d=dim\ P$. This allows one to define the Noether-Lefschetz locus $NL_{\beta}$ as the locus of quasi-smooth hypersurfaces of degree $\beta$ such that $i$ acting on the middle algebraic cohomology is not an isomorphism. In this paper we prove that, under some assumptions, if $dim P =2k+1$ and $k\beta-\beta_0=n\eta$ $(n\in\mathbb N)$, where $\eta$ is the class of a 0-regular ample divisor, and $\beta_0$ is the anticanonical class, then every irreducible component $V$ of the Noether-Lefschetz locus quasi-smooth hypersurfaces of degree $\beta$ satifies the bounds $n+1\leq codim\ V \leq h^{k-1,k+1}(X)$.