Moduli of surfaces of general type with a fibration by genus two curves
type: Item
instance of: scholarly article
title: Moduli of surfaces of general type with a fibration by genus two curves
zbMATH Open document ID: 0820.14026
DOI: 10.1007/BF01446659
author: Wolfgang K. Seiler
published in: Mathematische Annalen
publication date: 1995-09-17T00:00:00Z
full work available at URL: https://eudml.org/doc/165319
review text: 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 nonsingula…
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 759294
zbMATH Keywords: surfaces of general type
zbMATH Keywords: components of the moduli space
zbMATH Keywords: fibration by genus two curves
zbMATH Keywords: universal deformation
zbMATH Keywords: Euler number
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 759294