Bollettino dell'Unione Matematica Italiana

, Volume 11, Issue 1, pp 75–91 | Cite as

Canonical rings of Gorenstein stable Godeaux surfaces

Article

Abstract

Extending the description of canonical rings from Reid (J Fac Sci Univ Tokyo Sect IA Math 25(1):75–92, 1978) we show that every Gorenstein stable Godeaux surface with torsion of order at least 3 is smoothable.

Keywords

Stable surface Godeaux surface 

Mathematics Subject Classification

14J29 14J10 14H45 

Notes

Acknowledgements

The results in this paper are part of our exploration of Gorenstein stable surfaces with \(K_X^2 = 1\) carried out jointly with Rita Pardini. We would like to thank her for this enjoyable collaboration. S. Rollenske enjoyed several discussions with Stephen Coughlan and Roberto Pignatelli about canonical rings in general and Godeaux surfaces in particular. We also would like to thank Giancarlo Urzúa for comments. M. Franciosi is grateful for support by the PRIN project 2010S47ARA\(\_\)011 “Geometria delle Varietà Algebriche” of italian MIUR. S. Rollenske is grateful for support by the DFG via the Emmy Noether programme and partially SFB 701.

References

  1. 1.
    Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves, vol. I. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267. Springer, New York (1985)Google Scholar
  2. 2.
    Alexeev, V.: Higher-dimensional analogues of stable curves. In: International Congress of Mathematicians, vol. II, pp. 515–536. European Mathematical Society, Zürich (2006)Google Scholar
  3. 3.
    Barlow, R.: Some new surfaces with \(p_g=0\). Duke Math. J. 51(4), 889–904 (1984)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Barlow, R.: A simply connected surface of general type with \(p_g=0\). Invent. Math. 79(2), 293–301 (1985)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Catanese, F., Debarre, O.: Surfaces with \(K^2=2,\; p_g=1,\; q=0\). J. Reine Angew. Math. 395, 1–55 (1989)MathSciNetMATHGoogle Scholar
  6. 6.
    Catanese, F., Franciosi, M., Hulek, K., Reid, M.: Embeddings of curves and surfaces. Nagoya Math. J. 154, 185–220 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Coughlan, S.: Extending hyperelliptic K3 surfaces, and Godeaux surfaces with torsion \({\mathbb{Z}}/2\). J. Korean Math. Soc. (to appear)Google Scholar
  8. 8.
    Coughlan, S., Urzúa, G.: On \( {\mathbb{Z}}/3\)-Godeaux surfaces (2016). arXiv:1609.02177
  9. 9.
    Catanese, F., Pignatelli, R.: On simply connected Godeaux surfaces. In: Complex Analysis and Algebraic Geometry, pp. 117–153. de Gruyter, Berlin (2000)Google Scholar
  10. 10.
    Dolgachev, I., Werner, C.: A simply connected numerical Godeaux surface with ample canonical class. J. Algebraic Geom. 8(4), 737–764 (1999)MathSciNetMATHGoogle Scholar
  11. 11.
    Dolgachev, I., Werner, C.: Erratum to: “A simply connected numerical Godeaux surface with ample canonical class” [J. Algebraic Geom. 8 (1999), no. 4, 737–764; MR1703612 (2000h:14030)]. J. Algebraic Geom. 10(2), 397 (2001)Google Scholar
  12. 12.
    Franciosi, M., Pardini, R., Rollenske, S.: Computing invariants of semi-log-canonical surfaces. Math. Z. 280(3–4), 1107–1123 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Franciosi, M., Pardini, R., Rollenske, S.: Gorenstein stable Godeaux surfaces (2016). arXiv:1611.07184
  14. 14.
    Franciosi, M.: On the canonical ring of curves and surfaces. Manuscr. Math. 140(3–4), 573–596 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Franciosi, M., Tenni, E.: The canonical ring of a 3-connected curve. Rend. Lincei Mat. Appl. 25 (2014)Google Scholar
  16. 16.
    Inoue, M.: Some new surfaces of general type. Tokyo J. Math. 17(2), 295–319 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kollár, J.: Moduli of varieties of general type. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli: Volume II. Advanced Lectures in Mathematics, vol. 24, pp. 131–158. International Press (2012). arXiv:1008.0621
  18. 18.
    Kollár, J.: Singularities of the minimal model program. In: Cambridge Tracts in Mathematics, vol. 200. Cambridge University Press, Cambridge (2013) (With a collaboration of Sándor Kovács)Google Scholar
  19. 19.
    Kollár, J.: Moduli of varieties of general type (2016) (book in preparation)Google Scholar
  20. 20.
    Kollár, J., Shepherd-Barron, N.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Liu, W., Rollenske, S.: Pluricanonical maps of stable log surfaces. Adv. Math. 258, 69–126 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Liu, W., Rollenske, S.: Geography of Gorenstein stable log surfaces. Trans. Am. Math. Soc. 368(4), 2563–2588 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rana, J., Tevelev, J., Urzúa, G.: The Craighero–Gattazzo surface is simply-connected (2015). arXiv:1506.03529
  24. 24.
    Reid M.: Surfaces with \(p_{g}=0\), \(K^{2}=1\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25(1), 75–92 (1978)Google Scholar
  25. 25.
    Reid, M.: Gorenstein in codimension 4: the general structure theory. In: Algebraic Geometry in East Asia-Taipei 2011. Advanced Studies in Pure Mathematics, vol. 65, pp. 201–227. The Mathematical Society of Japan, Tokyo (2015)Google Scholar
  26. 26.
    Rollenske, S.: A new irreducible component of the moduli space of stable Godeaux surfaces. Manuscr. Math. 149(1–2), 117–130 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Werner, C.: A surface of general type with \(p_g=q=0\), \(K^2=1\). Manuscr. Math. 84(3–4), 327–341 (1994)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.FB 12/Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany

Personalised recommendations