  We take a sum $C_1+r C_2,\ r\in\mathbb Q$ 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 $C_1\cap C_2=\emptyset$. 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 $C_1+ r C_2$ 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 $\leq 9$, and many
non-reduced components for such degrees.
