Monodromies of hyperelliptic families of genus three curves
Let \(\phi :S\rightarrow \Delta\) be a proper surjective holomorphic map from a non-singular complex surface \(S\) to a small disk \(\Delta :=\{ t\in {\mathbb C}\mid | t| <\varepsilon \}\) such that \(\phi ^{-1}(t)\) is a non-singular curve of genus \(g\geq 2\) for each \(t\in \Delta ^*:=\Delta \backslash \{ 0\}\). Call \((\phi ,S,\Delta )\) a degeneration of curves of genus \(g\). If the curve \(\phi ^{-1}(t)\) is hyperelliptic for \(t\in \Delta ^*\), then \((\phi ,S,\Delta )\) is called a hyperelliptic family. In the paper the complete list of monodromies not realizable by hyperelliptic families of genus three is given. It is proved that all other monodromies of genus three are realized as monodromies of certain hyperelliptic families. In the proof an operation called ``the inverse of Horikawa's canonical resolution'' is used.
scientific article; zbMATH DE number 2158072