An asymptotic description of the Noether-Lefschetz components in toric varieties

We extend the definition of Noether-Leschetz components to quasi-smooth hypersurfaces in a projective simplicial toric variety of dimension 2k+1, and prove that asymptotically the components whose codimension is upper bounded by a suitable effective constant correspond to hypersurfaces with which one can associate a small degree k-dimensional subvariety. As a corollary we get an asymptotic characterization of the components with small codimension, generalizing Otwinowska's work for odd-dimensional projective spaces and Green and Voisin's for projective 3-space. Some tools developed in this paper are a generalization of Green's theorem for simplicial toric varieties, and an extension of the notion of artinian Gorenstein ideal for the Cox ring of a toric variety.