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On Schoen surfaces. (English) Zbl 1315.14053
Schoen surfaces are irregular surfaces of general type with $$p_g = 5$$, $$\chi = 2$$, $$K^2 = 16$$ first constructed by C. Schoen in [Int. J. Math. 18, No. 5, 585–612 (2007; Zbl 1118.14042)]. They are Lagrangian surfaces (i.e., Lagrangian subvarieties of their Albanese variety) and satisfy $$p_g = 2q - 3$$, which is the minimum possible $$p_g$$ with respect to $$q$$ for surfaces without irrational pencil of genus $$\geq 2$$.
The present article first reviews Schoen’s construction and some interesting properties of the resulting Schoen surfaces. In the second part of the article, the authors give a new approach to constructing Schoen surfaces and use this approach to obtain more information about a general Schoen surface $$S$$. In particular, they prove that the canonical map realizes $$S$$ as a double cover over a canonical surface with 40 even nodes, which is a complete intersection of a quadric and a quartic hypersurface in $$\mathbb{P}^4$$.

##### MSC:
 14J29 Surfaces of general type 32G05 Deformations of complex structures 14D06 Fibrations, degenerations in algebraic geometry 14J10 Families, moduli, classification: algebraic theory
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