Let $\hatX$ be a smooth connected subvariety of complex projective
space $\pn n$. The question was raised in \cite{CHS} of how to
characterize $\hatX$ if it admits a reducible hyperplane
section $\hatL$. In the case in which $\hatL$ is
the union of $r \geq 2$ smooth normal crossing divisors, each of
sectional genus zero, classification theorems were given for
$\dim \hatX \geq 5$ or $\dim X=4$ and $r=2$.
This paper restricts attention to the case of two divisors on a
threefold, whose sum is ample, and which meet transversely in a
smooth curve of genus at least $2$. A finiteness theorem and some
general results are proven, when the two divisors are in a
restricted class including $\pn 1$-bundles over curves of genus
less than two and surfaces with nef and big anticanonical bundle.
Next, we give results on the case of a projective threefold $\hatX$
with hyperplane section $\hatL$ that is the union of two transverse
divisors, each of which is either $\pn 2$, a Hirzebruch
surface $\eff_r$, or $\widetilde{\eff_2}$.