Abstract. The existence of a Kodaira fibration, i.e., of a fibration of a compact complex surface S onto a complex curve B which is a differentiable but not a holomorphic bundle, forces the geographical slope ν(S) = c 2 1 (S)/c 2 (S) to lie in the interval (2, 3). But up to now all the known examples had slope ν(S) ≤ 2 + 1/3. In this paper we consider a special class of surfaces admitting two such Kodaira fibrations, and we can construct many new examples, showing in particular that there are such fibrations attaining the slope ν(S) = 2 + 2/3. We are able to explicitly describe the moduli space of such class of surfaces, and we show the existence of Kodaira fibrations which yield rigid surfaces. We observe an interesting connection between the problem of the slope of Kodaira fibrations and a 'packing' problem for automorphisms of algebraic curves of genus ≥ 2.

In this paper we consider Gorenstein stable surfaces with KX2=1 and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with KX2=1; for pg=2 this leads to a rough stratification of the moduli space. Explicit non‐Gorenstein examples show that we need further techniques to understand all possible degenerations.

We explicitly describe the possible degenerations of a class of double Kodaira fibrations in the moduli space of stable surfaces. Using deformation theory we also show that under some assumptions we get a connected component of the moduli space of stable surfaces.