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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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