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Automorphisms of rational double points and moduli spaces of surfaces of general type. (English) Zbl 0615.14021

Let S be a complex projective nonsingular minimal surface of general type and let \({\mathcal M}(S)\) be the coarse moduli space of complex structures on the oriented topological 4-manifold underlying S. It is known that \({\mathcal M}(S)\) is a quasi-projective variety. The paper under review is the third of a series [(1) J. Differ. Geom. 19, 483-515 (1984; Zbl 0549.14012); (2) Algebraic Geometry, Open Problems, Proc. Conf., Ravello/Italy 1982, Lect. Notes Math. 997, 90-112 (1983; Zbl 0517.14011); (4) J. Differ. Geom. 24, 395-399 (1986)] devoted to the study of general properties of \({\mathcal M}(S)\). This study was carried out by using a test class of simply connected surfaces obtained by deforming bidouble covers of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\), i.e. Galois covers with group \(({\mathbb{Z}}/2)^ 2\). The interest in these bidouble covers is evident in view of the analogy with hyperelliptic curves in dimension \(1\).
Here the author studies in the large the deformations of these surfaces. A bidouble cover as above looks like the subvariety of the total space of the bundle \({\mathcal O}_{{\mathbb{P}}^ 1\times {\mathbb{P}}^ 1}(a,b)\oplus {\mathcal O}_{{\mathbb{P}}^ 1\times {\mathbb{P}}^ 1}(n,m)\) defined by equations \(z^ 2=f(x,y)\), \(w^ 2=g(x,y)\), where f and g are bihomogeneous forms of bidegrees (2a,2b) and (2n,2m) respectively. Let \({\mathcal N}_{(a,b),(n,m)}\) be the subset of the moduli space corresponding to smooth natural deformations; the author supplies a complete description of the closure of \({\mathcal N}_{(a,b),(n,m)}\), for \(a>2n\), \(m>2b\). This is achieved by exploiting the relations between deformations of bidouble covers of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) and degenerations of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) to normal surfaces with certain singularities, which the author calls ”1/2 rational double points” and studies in great detail.
Reviewer: A.Lanteri

MSC:

14J10 Families, moduli, classification: algebraic theory
14E20 Coverings in algebraic geometry
14D15 Formal methods and deformations in algebraic geometry
14J25 Special surfaces
14J17 Singularities of surfaces or higher-dimensional varieties
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References:

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