## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 25, Number 2 (2002), 139-150.

### Reducible hyperplane sections, II.

M.C. Beltrametti, K.A. Chandler, and A.J. Sommese

#### 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 $\widetilde{\eff_2}$.

#### Article information

**Source**

Kodai Math. J., Volume 25, Number 2 (2002), 139-150.

**Dates**

First available in Project Euclid: 17 December 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1071674437

**Digital Object Identifier**

doi:10.2996/kmj/1071674437

**Mathematical Reviews number (MathSciNet)**

MR2003h:14012

**Zentralblatt MATH identifier**

1048.14019

#### Citation

Beltrametti, M.C.; Chandler, K.A.; Sommese, A.J. Reducible hyperplane sections, II. Kodai Math. J. 25 (2002), no. 2, 139--150. doi:10.2996/kmj/1071674437. https://projecteuclid.org/euclid.kmj/1071674437