Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-20T12:35:13.908Z Has data issue: false hasContentIssue false
  • English
  • Français

The Slope of Surfaces with Albanese Dimension One

Published online by Cambridge University Press:  28 May 2018

STEFANO VIDUSSI
STEFANO VIDUSSI*
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Mendes Lopes and Pardini showed that minimal general type surfaces of Albanese dimension one have slopes K2/χ dense in the interval [2,8]. This result was completed to cover the admissible interval [2,9] by Roulleau and Urzua, who proved that surfaces with fundamental group equal to that of any curve of genus g ≥ 1 (in particular, having Albanese dimension one) give a set of slopes dense in [6,9]. In this note we provide a second construction that complements that of Mendes Lopes–Pardini, to recast a dense set of slopes in [8,9] for surfaces of Albanese dimension one. These surfaces arise as ramified double coverings of cyclic covers of the Cartwright–Steger surface.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 2 , September 2019 , pp. 355 - 360
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[BHPV04] Barth, W., Hulek, K., Peters, C. and Van de Ven, A. Compact complex surfaces. Second edition. Ergeb. Math. Grenzgeb. 3. Folge. A Series of Modern Surveys in Mathematics, 4 (Springer-Verlag, Berlin, 2004), xii+436 pp.Google Scholar
[Bea92] Beauville, A. Annulation du H 1 pour les fibrés en droites plats. Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math. vol. 1507 (Springer, Berlin, 1992), pp. 1–15.Google Scholar
[CKY17] Cartwright, D., Koziarz, V. and Yeung, S.-K. On the Cartwright–Steger surface. J. Alg. Geom. 26 (2017), no. 4, 655–689.Google Scholar
[CS10] Cartwright, D. and Steger, T. Enumeration of the 50 fake projective planes. C. R. Acad. Sci. Paris, Ser. 1, 348 (2010), 11–13.Google Scholar
[Cat03] Catanese, F. Fibred Kähler and quasi–projective groups. Special issue dedicated to Adriano Barlotti. Adv. Geom. suppl. (2003), 13–27.Google Scholar
[De08] Delzant, T. Trees, valuations and the Green–Lazarsfeld set. Geom. Funct. Anal. 18 (2008), no. 4, 1236–1250.Google Scholar
[De10] Delzant, T. L'invariant de Bieri Neumann Strebel des groupes fondamentaux des variétés kähleriennes. Math. Annalen 348 (2010), 119–125.Google Scholar
[DR17] Džambić, A. and Roulleau, X. On the cohomology of Stover surface. Exp. Math. 26 (2017), no. 4, 490–495.Google Scholar
[GL87] Green, M. and Lazarsfeld, R. Deformation theory, generic vanishing theorems and some conjectures of Enriques, Catanese and Beauville. Invent. Math. 90 (1987), no. 2, 389–407.Google Scholar
[GL91] Green, M. and Lazarsfeld, R. Higher obstruction to deforming cohomology groups of line bundles. J. Amer. Math. Soc. 4 (1991), no. 1, 87–103.Google Scholar
[Hi97] Hironaka, E. Alexander stratifications of character varieties. Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 555–583.Google Scholar
[MLP11] Mendes Lopes, M. and Pardini, R. The geography of irregular surfaces. In Current Developments in Algebraic Geometry (MSRI Publications, vol. 59 (2011)) 349–378.Google Scholar
[RU15] Roulleau, X. and Urzúa, G. Chern slopes of simply connected surfaces of general type are dense in [2, 3]. Ann. Math. 182 (2015), no. 1, 287–306.Google Scholar
[Si93] Simpson, C. Subspaces of moduli spaces of rank one local systems. Ann. Sci. Ecole Norm. Sup. 26 (1993), no. 3, 361–401.Google Scholar
[So84] Sommese, A. J. On the density of ratios of Chern numbers of algebraic surfaces. Math. Ann. 268 (1984), no. 2, 207–221.Google Scholar
[St14] Stover, M. Hurwitz ball quotients. Math. Z. 277 (2014), no. 1–2, 75–91.Google Scholar
[St18] Stover, M. On general type surfaces with q = 1 and c 2 = 3pg. Manuscripta Math. (to appear).Google Scholar