Compact moduli for certain Kodaira fibrations

Rollenske, Sönke

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 4, pp. 851-874.

We explicitly describe the possible degenerations of a class of double Kodaira fibrations in the moduli space of stable surfaces. Using deformation theory we also show that under some assumptions we get a connected component of the moduli space of stable surfaces.

[1] D. Abramovich and B. Hassett, Stable varieties with a twist, arXiv:0904.2797 (2009).

[2] V. Alexeev, Higher-dimensional analogues of stable curves, In: “International Congress of Mathematicians”, Vol. II, Zürich, 2006, Eur. Math. Soc, 515–536. | Zbl

[3] V. Alexeev, Limits of stable pairs, Pure Appl. Math. Q. 4 (2008), 767–783. | Zbl

[4] V. Alexeev and R. Pardini, Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, arXiv:0901.4431 (2009).

[5] D. Abramovich and A. Vistoli, Complete moduli for fibered surfaces, In: “Recent Progress in Intersection Theory”, Bologna, 1997, Trends Math., Birkhäuser Boston, Boston, MA, 2000, 1–31. | Zbl

[6] W. P. Barth, K. Hulek, C. A. M. Peters and A. Van De Ven, “Compact Complex Surfaces”, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 4, Springer-Verlag, Berlin, second edition, 2004. | Zbl

[7] F. Catanese and S. Rollenske, Double Kodaira fibrations, J. Reine Angew. Math. 628 (2009), 205–233. | Zbl

[8] B. Fantechi and M. Manetti, Obstruction calculus for functors of Artin rings. I, J. Algebra 202 (1998), 541–576. | Zbl

[9] D. Gieseker, Global moduli for surfaces of general type, Invent. Math. 43 (1977), 233–282. | EuDML | Zbl

[10] R. Hartshorne, “Algebraic Geometry”, Springer-Verlag, New York, 1977, Vol. 52, Graduate Texts in Mathematics. | Zbl

[11] B. Hassett, Stable log surfaces and limits of quartic plane curves, Manuscripta Math. 100 (1999), 469–487. | Zbl

[12] B. Hassett, Stable limits of log surfaces and Cohen-Macaulay singularities, J. Algebra 242 (2001), 225–235. | Zbl

[13] J. Harris and I. Morrison, “Moduli of Curves”, Graduate Texts in Mathematics, Vol. 87, Springer-Verlag, New York, 1998. | Zbl

[14] J. Kollár and S. Mori, “Birational Geometry of Algebraic Varieties”, Cambridge Tracts in Mathematics, Vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.

[15] S. J. Kovács, Young person’s guide to moduli of higher dimensional varieties, In: “Proceedings of the AMS Summer Research Institute held at the University of Washington”, Seattle, WA, July 25 - August 12, 2005, Proceedings of Symposia in Pure Mathematics. American Mathematical Society, 2005. | Zbl

[16] S. Kebekus and T. Peternell, A refinement of Stein factorization and deformations of surjective morphisms, Asian J. Math. 12 (2008), 365–389. | Zbl

[17] J. Kollár and N. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299–338. | EuDML | Zbl

[18] Z. Ran, Deformations of maps, In: “Algebraic Curves and Projective Geometry”, Trento, 1988, Vol. 1389, Lecture Notes in Math., Springer, Berlin, 1989, 246–253. | Zbl

[19] S. Tufféry, Déformations de courbes avec action de groupe, Forum Math. 5 (1993), 243–259. | EuDML | Zbl

[20] M. Van Opstall, Moduli of products of curves, Arch. Math. (Basel) 84 (2005), 148–154. | Zbl

[21] M. Van Opstall, Stable degenerations of surfaces isogenous to a product of curves, Proc. Amer. Math. Soc. 134 (2006) 2801–2806 (electronic). | Zbl