A result on bicanonical maps of surfaces of general type (Q1912844)

Pluricanonical maps of surfaces of general type have been studied for quite a long time. Problems are left open only for bicanonical maps with small \(K_S^2 (\leq 4)\), and for canonical maps. For bicanonical maps, the non-trivial cases are these surfaces with \(p_g(S)=0\) and \(K_S^2=3,4\). We have the following conjecture: If \(S\) is a minimal surface of general type with \(p_g(S)=0\) and \(K_S^2=3\) or 4, then the bicanonical map \(\Phi_{|2K_S|}\) has no fixed points. As always, in these cases the most difficult part is about the \((-2)\)-curves. Our first result then is the following theorem: Let \(S\) be a minimal surface of general type with \(p_g(S)=0\) and \(K_S^2=3\) or 4. Suppose \(C\) is a \((-2)\)-curve in \(S\). Then \(C\) cannot be an irreducible component of the fixed part of the bicanonical map \(\Phi_{|2K_S|}\).