Reducible hyperplane sections, II.

Abstract

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 $\widetilde{\eff_2}$.