Components of maximal dimension in the Noether-Lefschetz locus for Beilinson-Hodge cycles on open surfaces

The main subject of study in this paper is the Noether-Lefschetz locus for Beilinson-Hodge cycles on the complement of the union of hyperplane sections in a surface in ${\mathbb{P}}^3$. We give an analogue of the result by M.Green and C.Voisin on the maximal components in the Noether-Lefschetz locus.