From the introduction: Let \(\phi:S\to\Delta\) be a proper surjective holomorphic map from a complex surface \(S\) to the unit disk \(\Delta= \{t\in \mathbb C\mid |t|< 1\}\) such that \(\phi^{-1}(t)\) is a smooth curve of genus \(g\geq1\) for any \(t\in \Delta^* =\Delta-\{0\}\). We call \(\phi\) a degeneration of curves of genus \(g\) and call \(F =\phi^{-1}(0)\) the singular fiber. The most fundamental problem of degenerations of curves is to classify singular fibers, their monodromies and moduli points. After the monumental work of K. Kodaira [Ann. Math. (2) 71, 111–152 (1960; Zbl 0098.13004)] for elliptic curves, Y. Namikawa and K. Ueno [Manuscr. Math. 9, 143–186 (1973; Zbl 0263.14007)] classified singular fibers of genus two and determined their homological monodromies and the limits of their period matrices.
The main result of this paper is to classify singular fibers of genus three and determine their topological monodromies and the strata of their moduli points. More precisely, we explicitly determine the conjugacy class, arising from the usual monodromy action for each degeneration, of the mapping class group of a Riemann surface of genus three and determine the topological type of the stable curve which is the limit of the moduli map in Deligne-Mumford’s compactified moduli space \(\overline{{\mathbf M}}_3\) of genus three.
Our basic tools are two theorems that where studied and developed y several authors and settled by Y. Matsumoto and J. M. Montesinos-Amilibia [Bull. Am. Math. Soc., New Ser. 30, No. 1, 70–75 (1994; Zbl 0797.30036)].