Mathematische Annalen

, Volume 374, Issue 1–2, pp 1007–1032 | Cite as

Moduli of fibered surface pairs from twisted stable maps

  • Kenneth AscherEmail author
  • Dori Bejleri


In this paper, we use the theory of twisted stable maps to construct compactifications of the moduli space of pairs \((X \rightarrow C, S + F)\) where \(X \rightarrow C\) is a fibered surface, S is a sum of sections, F is a sum of marked fibers, and \((X,S+F)\) is a stable pair in the sense of the minimal model program. This generalizes the work of Abramovich–Vistoli, who compactified the moduli space of fibered surfaces with no marked fibers. Furthermore, we compare our compactification to Alexeev’s space of stable maps and the KSBA compactification. As an application, we describe the boundary of a compactification of the moduli space of elliptic surfaces.

Mathematics Subject Classification

14J10 14D23 



We thank our advisor Dan Abramovich for his constant support. We thank Valery Alexeev, Giovanni Inchiostro, Gabriele La Nave, and Dhruv Ranganathan for helpful suggestions.


  1. 1.
    Ascher, K., Bejleri, D.: Log canonical models of elliptic surfaces. Adv. Math. 320, 210–243 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ascher, K., Bejleri, D.: Moduli of weighted stable elliptic surfaces and invariance of log plurigenera. arXiv:1702.06107, (2017)
  3. 3.
    Abramovich, D.: Canonical models and stable reduction for plurifibered varieties. ArXiv Mathematics e-prints, (June 2002)Google Scholar
  4. 4.
    Abramovich, D., Chen, Q., Gillam, D., Huang, Y., Olsson, M., Satriano, M., Sun, S..: Logarithmic geometry and moduli. In Handbook of moduli. Vol. I, volume 24 of Adv. Lect. Math. (ALM), pp. 1–61. Int. Press, Somerville, MA, (2013)Google Scholar
  5. 5.
    Abramovich, D., Corti, A., Vistoli, A.: Twisted bundles and admissible covers. Commun. Algebra 31(8), 3547–3618 (2003). (Special issue in honor of Steven L. Kleiman (2003))MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Valery, A., Michael Guy, G.: Moduli of weighted stable maps and their gravitational descendants. J. Inst. Math. Jussieu 7(3), 425–456 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Alexeev, V.: Moduli spaces \(M_{g,n}(W)\) for surfaces. In: Higher-Dimensional Complex Varieties (Trento, 1994), pp. 1–22. de Gruyter, Berlin, (1996)Google Scholar
  8. 8.
    Abramovich, D., Olsson, M., Vistoli, A.: Twisted stable maps to tame Artin stacks. J. Algebraic Geom. 20(3), 399–477 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Abramovich, D., Vistoli, A.: Complete moduli for fibered surfaces. In: Recent Progress in Intersection Theory (Bologna, 1997), (1997)Google Scholar
  10. 10.
    Abramovich, D., Vistoli, A.: Compactifying the space of stable maps. J. Am. Math. Soc. 15(1), 27–75 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brunyate, A.: A Modular Compactification of the Space of Elliptic K3 Surfaces. PhD thesis, The University of Georgia, (2015)Google Scholar
  12. 12.
    Noah Giansiracusa and William Danny Gillam: On Kapranov’s description of \({\overline{M}}_{0, n}\) as a Chow quotient. Turkish J. Math. 38(4), 625–648 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kato, F.: Log smooth deformation and moduli of log smooth curves. Int. J. Math. 11(2), 215–232 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    La Nave, G.: Explicit stable models of elliptic surfaces with sections. arXiv:math/0205035
  15. 15.
    Miranda, R.: The basic theory of elliptic surfaces. Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research]. ETS Editrice, Pisa, (1989)Google Scholar
  16. 16.
    Olsson, M.C.: (Log) twisted curves. Compos. Math. 143(2), 476–494 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Patakfalvi, Z.: Fibered stable varieties. Trans. Amer. Math. Soc. 368(3), 1837–1869 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate texts in Mathematics, vol. 106, 2nd edn. Springer, Dordrecht (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Brown UniversityProvidenceUSA

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