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Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 5, Pages 123–170 (Mi izv505)  

This article is cited in 12 scientific papers (total in 12 papers)

Old and new examples of surfaces of general type with $p_g=0$

Vik. S. Kulikov

Steklov Mathematical Institute, Russian Academy of Sciences

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 $k\geqslant 2$ 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 $(\mathbb Z/3\mathbb Z)^3\subset\operatorname{Tors}(X)$. 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 $\operatorname{Tors}(S)\simeq(\mathbb Z/2\mathbb Z)^3$ (and hence belongs to the family of Campedelli surfaces), that is, the corresponding statement in [9], [4], and [1, p. 237], about the torsion group of the Burniat surface $S$ with $K_S^2=2$ is not correct.


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English version:
Izvestiya: Mathematics, 2004, 68:5, 965–1008

Bibliographic databases:

UDC: 512.7
MSC: 14J25, 14J10, 32J15, 14J17
Received: 13.04.2004

Citation: Vik. S. Kulikov, “Old and new examples of surfaces of general type with $p_g=0$”, Izv. RAN. Ser. Mat., 68:5 (2004), 123–170; Izv. Math., 68:5 (2004), 965–1008

Citation in format AMSBIB
\by Vik.~S.~Kulikov
\paper Old and new examples of surfaces of general type with $p_g=0$
\jour Izv. RAN. Ser. Mat.
\yr 2004
\vol 68
\issue 5
\pages 123--170
\jour Izv. Math.
\yr 2004
\vol 68
\issue 5
\pages 965--1008

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    This publication is cited in the following articles:
    1. Vik. S. Kulikov, V. M. Kharlamov, “Surfaces with DIF$\ne$DEF real structures”, Izv. Math., 70:4 (2006), 769–807  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Calabri A., Mendes Lopes M., Pardini R., “Involutions on numerical Campedelli surfaces”, Tohoku Math. J. (2), 60:1 (2008), 1–22  crossref  mathscinet  isi  scopus
    3. Bauer I., Catanese F., “Burniat surfaces II: secondary Burniat surfaces form three connected components of the moduli space”, Inventiones Mathematicae, 180:3 (2010), 559–588  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Bauer I., Catanese F., Grunewald F., Pignatelli R., “Quotients of Products of Curves, New Surfaces with P(G)=0 and their Fundamental Groups”, Am. J. Math., 134:4 (2012), 993–1049  crossref  mathscinet  zmath  isi  scopus
    5. Bauer I., Catanese F., “Burniat Surfaces III: Deformations of Automorphisms and Extended Burniat Surfaces”, Doc. Math., 18 (2013), 1089–1136  mathscinet  zmath  isi
    6. Chan Tsz On Mario, Coughlan S., “Kulikov Surfaces Form a Connected Component of the Moduli Space”, Nagoya Math. J., 210 (2013), 1–27  crossref  mathscinet  zmath  isi  scopus
    7. Vik. S. Kulikov, V. M. Kharlamov, “On numerically pluricanonical cyclic coverings”, Izv. Math., 78:5 (2014), 986–1005  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. Neves J., Pignatelli R., “Unprojection and Deformations of Tertiary Burniat Surfaces”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 13:1 (2014), 225–254  mathscinet  zmath  isi
    9. Coughlan S., “Enumerating Exceptional Collections of Line Bundles on Some Surfaces of General Type”, Doc. Math., 20 (2015), 1255–1291  mathscinet  zmath  isi
    10. Dimca A., “Hyperplane Arrangements: An Introduction”, Hyperplane Arrangements: An Introduction, Universitext, Springer, 2017, 1–200  crossref  mathscinet  isi
    11. Cancian N., Frapporti D., “On Semi-Isogenous Mixed Surfaces”, Math. Nachr., 291:2-3 (2018), 264–283  crossref  mathscinet  zmath  isi  scopus
    12. Vik. S. Kulikov, “On divisors of small canonical degree on Godeaux surfaces”, Sb. Math., 209:8 (2018), 1155–1163  mathnet  crossref  crossref  adsnasa  isi  elib
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