Bruzzo, UgoMontoya, William D.2019-03-192019-03-192019-03-19https://openscience.sissa.it/handle/1963/35331We extend the definition of Noether-Leschetz components to quasi-smooth hyper- surfaces in a projective toric variety PΣ2k+1 having orbifold singularities, and prove that asymptoticaly the components whose codimension is bounded from above are made of hy- persurfaces containing a small degree k-dimensional subvariety. As a corollary we get an asymptotic characterization of the components with small codimension, generalizing the work of Otwinowska for P2k+1 = P2k+1 and Green and Voisin for P2k+1 = P3. Some tools that are developed in the paper are a generalization of Macaulay’s theorem for Fano, irreducible normal varieties with rational singularieties, satisfying a suitable additional condition, and an extension of the notion of Gorenstein ideal for normal varieties with finetely generated Cox ring.enNoether-Lefschetz locusPicard numbertoric varietiesPreprint