Standard isotrivial fibrations with \(p_g=q=1\) (Q1018402)

Several authors have studied surfaces of general type with \(p_g=q=1\), but these surfaces are still not completely understood. A surface \(S\) is a \textit{standard isotrivial fibration} if there exists a finite group \(G\) acting faithfully on two smooth projective curves \(C\) and \(F\) so that \(S\) is isomorphic to the minimal desingularization of \(T:=(C\times F)/G\). The case where \(T\) is smooth was classified by the author and \textit{G. Carnovale} [Commun. Algebra 36, No. 6, 2023--2053 (2008; Zbl 1147.14017), Adv. Geom. 9, No. 2, 233--256 (2009; Zbl 1190.14036)]; in this paper the case where \(T\) has only rational double points as singularities is considered; the general case is treated by the author and \textit{E. Mistretta} [``Standard isotrivial fibrations with \(p_g=q=1\). II'', J. Pure Appl. Algebra, in press, cf. \url{arXiv:0805.1424}]. Using the classification of finite groups acting on Riemann surfaces, it is shown that \(|G|\leq 168\). The group database of the computer algebra program GAP4 is used when working with groups of big order. The main result of this paper is a classification list of all cases that occur. In particular, new examples of surfaces of general type with \(p_g=q=1\) and \(K^2=4,6\) are obtained (these are different from the examples given by \textit{F. Catanese} [Contemp. Math. 241, 97--120 (1999; Zbl 0964.14012 )], \textit{H. Ishida} [Manuscr. Math. 118, No. 4, 467--483 (2005; Zbl 1092.14048 )] and the reviewer [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 1, 81--102 (2007; Zbl 1180.14040)]).