Special components of Noether-Lefschetz loci

We take a sum C-1 + rC(2), r is an element of Q of a line C-1 and a complete intersection curve C-2 of type (3, 3) inside a smooth surface of degree 8 and with C-1 boolean AND C-2 = empty set. We gather evidences to the fact that for all except a finite number of r, the Noether-Lefschetz loci attached to the cohomology classes of C-1 + rC(2) are distinct 31 codimensional subvarieties intersecting each other in a 32 codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is 35, and hence, we provide a conjectural description of a counterexample to a conjecture of J. Harris. The methods used in this paper also produce in a rigorous way an infinite number of general components passing through the point representing the Fermat surface of degree <= 9, and many non-reduced components for such degrees.