We classify singular fibers of curves of genus three and determine their topological monodromies and the strata of their moduli points. Introduction.Let φ : S be a proper surjective holomorphic map from a complex surface S to the unit disk1 for any t A* = {0}. We call φ a degeneration of curves of genus g and call F = φ 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 Kodaira [Ko] for elliptic curves, Namikawa-Ueno [NU1], [NU2] classified singular fibers of genus two and determined their homological monodromies and the limits of their period matrices.The main result of this paper (Theorem 4.3) 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 M3 of genus three.Our basic tools are the following Theorems (A) and (B), which was first studied by Nielsen [Ni2], developed by [Cl], [Im], [ES], [ST], [AMO], etc., and finally settled by Matsumoto-Montesinos [MM1], [MM2]:(A) ([MM2, Theorem 1]) The conjugacy class, realized as the topological monodromy of a degeneration of curves, of the mapping class group of a Riemann surface of genus g 2 is represented by a pseudo-periodic map of negative type. Conversely, any conjugacy class of pseudo-periodic map of negative type is realized as the topological monodromy of a certain degeneration of curves.(B) ([MM2, Theorem 2]) The conjugacy class of a pseudo-periodic map f : Σ g Σ g of negative type of a smooth curve Σ g of genus g is determined by the following data: An admissible system of cut curves C = \J C i on Σ g , the action of f on the oriented graph GC induced by C, the screw numbers of f around each annulus of C i and the valency data of

Let Σg be an oriented connected real two dimensional manifold of genus g without boundary. For periodic homeomorphisms of Σg that commute with a hyperelliptic involution, we give a method to obtain their presentations by Dehn twists.

Abstract. We classify the periodic monodromies which are realized as the monodromies of hyperelliptic families.