On Schoen surfaces. (English) Zbl 1315.14053

Ciliberto, Ciro; Mendes Lopes, Margarida; Roulleau, Xavier

Comment. Math. Helv. 90, No. 1, 59-74 (2015).

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 \(\mathbb{P}^4\).

Reviewer: Florian Schrack (Bayreuth)

MSC:

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

Keywords:

irregular surfaces; Lagrangian surfaces; deformations of surfaces; canonical maps