Monodromies of hyperelliptic families of genus three curves (Q2758470)

Given a proper surjective map \(\varphi: S \rightarrow \Delta\) from a complex surface \(S\) to an open disk \(\Delta\) such that \(\varphi^{-1}(t)\) is a smooth curve of genus \(g\geq 2\) for all \(t\in \Delta \smallsetminus \{0\}\), the triple \((\varphi,S,\Delta)\) is called a degeneration of curves. If all the inverse images are hyperelliptic curves the family is called hyperelliptic. Two degenerations are topologically equivalent if there is an orientation preserving homeomorphism between them that commutes with the projections. Define \(T_{g}\) to be the corresponding set of equivalence classes where \(g\) denotes the (fixed) genus. Each element of \(T_{g}\) determines a fixed monodromy as the conjugacy class of a pseudo-periodic map of the surface of genus \(g\). This paper classifies the monodromies of degenerations of genus three which are realized as monodromies of elliptic families, as well as those that cannot be so represented.