The moduli space of Catanese-Ciliberto-Ishida surfaces
Let \(S\) be a smooth minimal surface of general type with geometric genus and irregularity \(p_g=q=1\) and self-intersection of the canonical divisor \(K^2=3\). The Albanese map \(a:S\rightarrow E\) is a surjection to an elliptic curve \(E\) with smooth general fibre. \textit{F. Catanese} and \textit{C. Ciliberto} [J. Algebr. Geom. 2, No. 3, 389--411 (1993; Zbl 0791.14015)] have shown that the genus \(g\) of this fibre is \(2\) or \(3\). Moreover, if \(g=3\), then \(V:=a_*K_S\otimes_{\mathcal O_E} K_E^{-1}\) is a locally free sheaf of rank \(3\) and the map \(\phi:S\rightarrow {\mathbb{P}}_E(V)\) is a morphism. Later \textit{H. Ishida} [Tohoku Math. J. (2) 58, No. 1, 33--69 (2006; Zbl 1112.14012)] considered the case where \(\phi\) is an embedding and the Albanese map has only one singular fibre. In the paper under review these are called Catanese-Ciliberto-Ishida surfaces. Examples of these surfaces are constructed and their moduli space is determined.
surface of general type with \(p_g=q=1\)
scientific article; zbMATH DE number 6159569