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Explicit Schoen surfaces. (English) Zbl 1428.14071
In this extremely interesting paper, the authors construct in an explicit way the \(4\)-dimensional family of Schoen surfaces by computing equations for their canonical images.
The main result of the paper can be summed up as follows.
Main Result. Let \(I_{4}\) be the Igusa quartic in \(\mathbb{P}^{4}_{\mathbb{C}}\). There exists a quadric on four parameteres \(Q_{a,b,c,d}\) such that for generic values of these parameters the surface \[X_{40} := I_{4} \cap Q_{a,b,c,d}\] has exactly \(40\) nodes. The nodes are \(2\)-divisible in the Picard group, and the double cover \(S \rightarrow X_{40}\) ramified over the nodes is a Schoen surface (i.e., \(K_{s}^{2} = 2e(S) =16\), \(q(S)=4\), \(p_{g}(S) = 5\)).
The authors show explicitly that the surface \(S\) is not covered by the bidisk \(\mathbb{H}\times \mathbb{H}\).
In the remaining part of the paper the authors study a certain particular surface \(\overline{S}\) obtained as the double cover of a particular \(40\)-nodal and degree \(8\) complete intersection surface with a large group of symmetries. Let us denote by \(\overline{X_{40}} \subset \mathbb{P}^{4}_{\mathbb{C}}\) the intersection of the following quadric and quartic: \[5(x^{2}+y^{2}+z^{2}+w^{2}+t^{2}) -7(x+y+z+w+t)^{2} = 0,\] \[4(x^{4}+y^{4}+z^{4}+w^{4}+t^{4}+h^{4}) - (x^{2} +y^{2}+z^{2}+w^{2}+t^{2}+h^{2})^{2} = 0,\] where \(h = -(x+y+z+w+t)\). The surface \(\overline{X_{40}}\) has exactly \(40\) nodes. Let \(\overline{S} \rightarrow \overline{X_{40}}\) be the double cover branched over the \(40\) nodes and let \(\hat{X}_{40}\) be the minimal resolution of \(\overline{X_{40}}\). Using the symmetries (i.e., the permutation group \(S_{5}\) is a subgroup of the automorphism group of \(\overline{X_{40}}\)) the authors are able to show that \(\overline{S}\) and \(\hat{X}_{40}\) have maximal Picard numbers equal to \(12\) and \(52\), respectively.
Due to the fact that the review should be rather concise, the reviewer warmly suggest to consult details of the paper with the emphasis of very nice constructions.
MSC:
14J29 Surfaces of general type
14J28 \(K3\) surfaces and Enriques surfaces
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