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Old examples and a new example of surfaces of general type with \(p_ g=0\). (English. Russian original) Zbl 1073.14055

Izv. Math. 68, No. 5, 965-1008 (2004); translation from Izv. Ross. Akad. Nauk Ser. Mat. 68, No. 5, 123-170 (2004).
Author’s abstract: We investigate surfaces of general type with geometric genus \(p_g = 0\) which may be given as Galois coverings of the projective plane branched over an arrangement of lines with Galois group G =\(( {\mathbb Z}/{q \mathbb Z})^k\), where \(k\geq 2\) and \(q\) is a prime. Examples of such coverings include the classical Godeaux surface, Campedelli surfaces, Burniat surfaces, and a new surface X with invariants \(K_X^2 = 6\) and \(( {\mathbb Z}/{3 \mathbb Z})\subset \text{Tors}(X)\). We prove that the automorphism group of a generic surface of Campedelli type is isomorphic to \(( {\mathbb Z}/ {2 \mathbb Z})^3\). We describe the irreducible components of the moduli space containing the Burniat surfaces. We also show that the Burniat surface \(S\) with \(K_S^2=2\) has torsion group \(\text{Tors}(S)\simeq ( {\mathbb Z}/{2 \mathbb Z})^3\) (and hence belongs to the family of Campedelli surfaces), that is, the corresponding statement in I. Dolgachev [“Algebraic surfaces with \(p_g=q=0\)”, Proc. CIME Cortona 1977, 97–215 (Liguori, Napoli) (1981)], C. A. M. Peters [Nagoya Math. J. 66, 109–119 (1977; Zbl 0329.14019)] and W. Barth, C. Peters and A. Van de Ven [“Compact complex surfaces”. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Band 4. (Berlin etc.: Springer-Verlag) (1984; Zbl 0718.14023)] about the torsion group of the Burniat surface with \(K_S^2=2\) is not correct.

MSC:

14J29 Surfaces of general type
14J25 Special surfaces
14J10 Families, moduli, classification: algebraic theory
32J15 Compact complex surfaces
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