On explicit constructions of rational elliptic surfaces with multiple fibers (Q1293337)

In earlier work the author [\textit{Y. Fujimoto}, Publ. Res. Inst. Math. Sci. 26, No. 1, 1-13 (1990; Zbl 0729.14027)] shows how rational elliptic surfaces with a multiple fibre are constructed from cubic curves on \({\mathbb{P}}^2\). Such so called Halphen pencils have also been considered by \textit{M. Nagata} [``On rational surfaces. I. II'', Mem. Coll. Sci., Univ. Kyoto, Ser. A 32, 351-370 and 33, 271-293 (1960; Zbl 0100.16703 and Zbl 0100.16801)] and \textit{F. R. Cossec} and \textit{I. V. Dolgachev} [``Enriques surfaces. I'', Prog. Math. 76 (1989; Zbl 0665.14017)]. In the current paper, the author creates explicitly new examples of Halphen pencils by composition of blow up and down of disjoint \((-1)\)-curves on a given Halphen pencil.