Elliptic surfaces with an ample divisor of genus two (Q1183111)

The article complements the classification of smooth complex algebraic surfaces \(S\) with an ample divisor of arithmetic genus 2 undertaken by \textit{M. Beltrametti}, \textit{A. Lanteri} and \textit{M. Palleschi} [Ark. Mat. 25, 189-210 (1987; Zbl 0645.14015)] by settling the case when \(S\) has Kodaira dimension 1. Such surfaces have a (unique) elliptic fibration \(\varphi:S\to C\) and, after shortening (and correcting) the list from the cited paper of possible types of such fibrations, the author gives a precise description of \(S\) when \(C\) is rational and \(\chi({\mathcal O}_ S)=0\). His main tool is a lemma by \textit{T. Katsura} and \textit{K. Ueno} [Math. Ann. 272, 291-330 (1985; Zbl 0553.14019)] which imposes restrictions on the multiplicities of the components of the singular fibers of \(\varphi\). The article ends with the construction of such a surface.