Surfaces with \(p_g=q=1\), \(K^2=6\) and non-birational bicanonical maps
Thanks to the efforts of several authors the behaviour of the bicanonical map of surfaces of general type is well understood, apart from a very few exceptions of surfaces with small numerical invariants. The present paper addresses one of these exceptions: minimal surfaces of general type with $p_g=q=1$ and $K^2=6$. It is shown that the bicanonical map of such surfaces is either birational or has degree 2. Note that examples for this last case were constructed by \textit{C. Rito} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 1, 81--102 (2007; Zbl 1180.14040)] and as such this bound is effective. \par By Reider's theorem it is known that the bicanonical map of these surfaces is a morphism and as such the possible degrees of the bicanonical map would be 1, 2, 3 or 4. The result is obtained by a long and detailed analysis excluding the possibility of degree bigger than 2.
scientific article; zbMATH DE number 7043606
scientific article