Open problems

  • Ciro Ciliberto
  • Th. Dedieu
  • F. Flamini
  • R. Pardini
  • C. Galati
  • S. Rollenske
Article
  • 35 Downloads

Abstract

A problem session, has been held during the workshop “Birational geometry of surfaces” which took place at the Department of Mathematics of the University of Rome “Tor Vergata”, in January, 11–15, 2016. In the following paper, we gather problems and questions that have been proposed and discussed during the event.

References

  1. 1.
    Albano, A., Pirola, G.P.: Dihedral monodromy and Xiao fibrations. Ann. Mat. Pura Appl. 195(4), 1255–1268 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barja, M.A., González-Alonso, V., Naranjo, J.C.: Xiao’s conjecture for general fibred surfaces. J. Reine Angew. Math. arXiv:1401.7502.  https://doi.org/10.1515/crelle-2015-0080 (2018, to appear)
  3. 3.
    Barja, M.A., Stoppino, L.: Linear stability of projected canonical curves with applications to the slope of fibred surfaces. J. Math. Soc. Jpn. 60(1), 171–192 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Barja, M.A., Zucconi, F.: On the slope of fibred surfaces. Nagoya Math. J. 164, 103–131 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Beauville, A.: L’inégalité \(p_g\ge 2q+4\) pour les surfaces de type général. Bull. Soc. Math. Fr. 110, 343–346 (1982)Google Scholar
  6. 6.
    Catanese, F., Dettweiler, M.: Answer to a question by Fujita on variation of Hodge structures. arXiv:1311.3232
  7. 7.
    Fujita, T.: On Kähler fiber spaces over curves. J. Math. Soc. Jpn. 30(4), 779–794 (1978)CrossRefMATHGoogle Scholar
  8. 8.
    González-Alonso, V., Stoppino, L., Torelli, S.: On the rank of the flat unitary factor of the Hodge bundle. arXiv:1709.05670
  9. 9.
    Konno, K.: On the irregularity of special non-canonical surfaces. Publ. RIMS Kyoto Univ. 30, 671–688 (1994)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lu, X., Zuo, K.: On the slope conjecture of Barja and Stoppino for fibred surfaces. arXiv:1504.06276
  11. 11.
    Pirola, G.P.: On a conjecture of Xiao. J. Reine Angew. Math. 431, 75–89 (1992)MathSciNetMATHGoogle Scholar
  12. 12.
    Xiao, G.: Fibred algebraic surfaces with low slope. Math. Ann. 276, 449–466 (1987)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Alexeev, V.: Boundedness and \({K}^2\) for log surfaces. Int. J. Math. 5(6), 779–810 (1994)CrossRefMATHGoogle Scholar
  14. 14.
    Franciosi, M., Pardini, R., Rollenske, S.: Gorenstein stable surfaces with \(K^2_X=1\) and \(p_g>0\). Math. Nachr. 290(5–6), 794–814 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    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, Vienna (2012). arXiv:1008.0621
  16. 16.
    Kollár, J.: Moduli of varieties of general type (2010). arXiv:1008.0621v1
  17. 17.
    Kollár, J., Shepherd-Barron, N.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lazarsfeld, R.: Positivity in algebraic geometry I. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. Classical Setting: Line Bundles and Linear Series, vol. 48. Springer, Berlin (2004)Google Scholar
  19. 19.
    Rana, J., Urzúa, G.: Optimal bounds for t-singularities in stable surfaces. arXiv:1708.02278
  20. 20.
    Accola, R.D.M.: On Castelnuovo’s inequality for algebraic curves. I. Trans. Am. Math. Soc. 251, 357–373 (1979)MathSciNetMATHGoogle Scholar
  21. 21.
    Aprodu, M.: Remarks on syzygies of \(d\)-gonal curves. Math. Res. Lett. 12, 387–400 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Castelnuovo, G.: Sulle curve che posseggono una infinità continua di corrispondenze algebriche. Scritti matematici offerti ad Enrico d’Ovidio, pp. 164–174 (1918)Google Scholar
  23. 23.
    Coppens, M., Martens, G.: Secant spaces and Clifford’s theorem. Compos. Math. 78, 193–212 (1991)MathSciNetMATHGoogle Scholar
  24. 24.
    Eisenbud, D., Lange, H., Martens, G., Schreyer, F.-O.: The Clifford dimension of a projective curve. Compos. Math. 72, 173–204 (1989)MathSciNetMATHGoogle Scholar
  25. 25.
    Green, M.L.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125–171 (1984)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Green, M.L., Lazarsfeld, R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83, 73–90 (1986)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Knutsen, A.L.: Exceptional curves on Del Pezzo surfaces. Math. Nachr. 256, 58–81 (2003)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Knutsen, A.L.: On two conjectures for curves on \(K3\) surfaces. Int. J. Math. 20, 1547–1560 (2009)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Knutsen, A.L., Lopez, A.F.: Brill–Noether theory of curves on Enriques surfaces, II. The Clifford index. Manuscr. Math. 147, 193–237 (2015)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Martens, G.: On dimension theorems of the varieties of special divisors on a curve. Math. Ann. 267, 279–288 (1984)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Faenzi, D., Polizzi, F., Vallès, J.: Triple planes with \(p_g=q=0\). Trans. Am. Math. Soc. arXiv:1605.02102 (2018, to appear)
  32. 32.
    When is a general projection of \(d^2\) points in \({\mathbb{P}}^3\) a complete intersection? MathOverflow question 67265, asked by F. Polizzi (2011)Google Scholar
  33. 33.
    Caucci, F., Cho, Y., Rizzi, L.: On dominant maps from a very general complete intersection surface in \({\mathbb{P}}^4\). Le Matematiche 72(2), 183–194 (2017)MathSciNetGoogle Scholar
  34. 34.
    Diaz, S.: A bound on the dimensions of complete subvarieties of \(M_g\). Duke Math. J. 51(2), 405–408 (1984)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lee, Y., Pirola, G.: On subfields of the function field of a general surface in \({\mathbb{P}}^3\). Int. Math. Res. Not. 24, 13245–13259 (2015)CrossRefMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2018

Authors and Affiliations

  • Ciro Ciliberto
    • 1
  • Th. Dedieu
    • 2
  • F. Flamini
    • 1
  • R. Pardini
    • 3
  • C. Galati
    • 4
  • S. Rollenske
    • 5
  1. 1.Università degli Studi di Roma Tor VergataRomeItaly
  2. 2.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France
  3. 3.Università di PisaPisaItaly
  4. 4.Università della CalabriaArcavacata, di Rende (CS)Italy
  5. 5.Philipps-Universität MarburgMarburgGermany

Personalised recommendations