Moduli of surfaces of general type with a fibration by genus two curves
In this paper, I study those components of the moduli space of surfaces \(X\) of general type which parametrize surfaces that admit a fibration by genus two curves over a nonsingular curve \(C\) of arbitrary genus, and whose canonical model is a covering of a ruled surface; this generalizes Horikawa's papers on the case \(C = \mathbb{P}^ 1\). For technical reasons, only surfaces whose Euler number is sufficiently big with respect to the genus \(g\) of \(C\) can be considered. The paper starts with a review of known properties of surfaces with a genus two fibration and with criteria for the existence of surfaces with given invariants. Once this existence is established, the surfaces can be deformed; the first main result of the paper says that the base space of the universal deformation is nonsingular for surfaces whose ramification divisor over the ruled surface is not too singular, and it has dimension \(7n\), where \(n = \omega^ 2_ X - \chi ({\mathcal O}_ X) + 7 - 7g\). Next I show that every surface of the type considered here can be deformed into one whose ramification divisor has only those mild singularities allowed in the theorem, hence the theorem holds for all generic surfaces. -- The rest of the paper deals with the question under what conditions two surfaces can be deformed into each other, this is, how many components of the moduli space parametrize surfaces of the type considered here. It turns out that these surfaces always form connected components of their own, and if the Euler number of the surface is sufficiently big with respect to the genus of the base curve, there are one or two such components, both of which are irreducible. The first one, which always occurs, parametrizes surfaces with a connected ramification divisor, the other one, which occurs if and only if the self-intersection number of the canonical divisor is divisible by eight, parametrizes surfaces whose ramification divisor is disconnected. Both components are generically nonsingular, hence their dimensions are the same as those of the base spaces of the universal deformation, that is \(7n\). -- For smaller values of the Euler number, the same components still occur, but there might be additional ones.
scientific article; zbMATH DE number 759294
scientific article