On surfaces with p 𝑔 = q = 1 and non-ruled bicanonical involution

Carlos Rito

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 1, page 81-102
  • ISSN: 0391-173X

Abstract

top
This paper classifies surfaces S of general type with p g = q = 1 having an involution i such that S / i has non-negative Kodaira dimension and that the bicanonical map of S factors through the double cover induced by i . It is shown that S / i is regular and either: a) the Albanese fibration of S is of genus 2 or b) S has no genus 2 fibration and S / i is birational to a K 3 surface. For case a) a list of possibilities and examples are given. An example for case b) with K 2 = 6 is also constructed.

How to cite

top

Rito, Carlos. "On surfaces with p$_{\textit {g}}$ = q = 1 and non-ruled bicanonical involution." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.1 (2007): 81-102. <http://eudml.org/doc/272263>.

@article{Rito2007,
abstract = {This paper classifies surfaces $S$ of general type with $p_g=q=1$ having an involution $i$ such that $S/i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i.$ It is shown that $S/i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S/i$ is birational to a $K3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^2=6$ is also constructed.},
author = {Rito, Carlos},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {involution; bicanonical map; irregular surface},
language = {eng},
number = {1},
pages = {81-102},
publisher = {Scuola Normale Superiore, Pisa},
title = {On surfaces with p$_\{\textit \{g\}\}$ = q = 1 and non-ruled bicanonical involution},
url = {http://eudml.org/doc/272263},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Rito, Carlos
TI - On surfaces with p$_{\textit {g}}$ = q = 1 and non-ruled bicanonical involution
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 1
SP - 81
EP - 102
AB - This paper classifies surfaces $S$ of general type with $p_g=q=1$ having an involution $i$ such that $S/i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i.$ It is shown that $S/i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S/i$ is birational to a $K3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^2=6$ is also constructed.
LA - eng
KW - involution; bicanonical map; irregular surface
UR - http://eudml.org/doc/272263
ER -

References

top
  1. [1] A. Beauville, “Surfaces Algébriques Complexes”, Astérisque 54, 1978. Zbl0394.14014
  2. [2] E. Bombieri, Canonical models of surfaces of general type, Publ. Math. Inst. Hautes Étud. Sci. 42 (1972), 171–219. Zbl0259.14005MR318163
  3. [3] G. Borrelli, On the classification of surfaces of general type with non-birational bicanonical map and Du Val double planes, preprint, math.AG/0312351. Zbl1132.14036MR2357686
  4. [4] W. Barth, C. Peters and A. Van de Ven, “Compact Complex Surfaces”, Ergebn. der Math. 3, Folge, Band 4, Springer-Verlag, Berlin, 1984. Zbl1036.14016MR749574
  5. [5] F. Catanese, On a class of surfaces of general type, In: “Algebraic Surfaces”, CIME, Liguori, 1981, 269–284. 
  6. [6] F. Catanese, Singular bidouble covers and the construction of interesting algebraic surfaces, Amer. Math. Soc. Contemp. Math.241 (1999), 97–120. Zbl0964.14012MR1718139
  7. [7] F. Catanese and C. Ciliberto, Surfaces with p g = q = 1 , In: Sympos. Math., Vol. 32, Academic Press, 1991, 49–79. Zbl0828.14024MR1273372
  8. [8] F. Catanese, C. Ciliberto and M. Mendes Lopes, On the classification of irregular surfaces of general type with nonbirational bicanonical map, Trans. Amer. Math. Soc.350 (1988), 275–308. Zbl0889.14019MR1422597
  9. [9] A. Calabri, C. Ciliberto and M. Mendes LopesNumerical Godeaux surfaces with an involution, Trans. Amer. Math. Soc., to appear. Zbl1124.14036MR2272143
  10. [10] C. Ciliberto, P. Francia and M. Mendes Lopes, Remarks on the bicanonical map for surfaces of general type, Math. Z.224 (1997), 137–166. Zbl0871.14011MR1427708
  11. [11] C. Ciliberto, The bicanonical map for surfaces of general type, Proc. Sympos. Pure Math.62 (1997), 57–84. Zbl0929.14022MR1492518
  12. [12] C. Ciliberto and M. Mendes Lopes, On surfaces with p g = q = 2 and non-birational bicanonical map, Adv. Geom.2 (2002), 281–300. Zbl1027.14018MR1924760
  13. [13] C. Ciliberto and M. Mendes Lopes, On surfaces with p g = 2 , q = 1 and non-birational bicanonical map, In: “Algebraic Geometry”, Beltrametti, Mauro C. et al. (eds.), a volume in memory of Paolo Francia, de Gruyter, Berlin, 2002, 117–126. Zbl1050.14028MR1954060
  14. [14] O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. Fr.110 (1982), 319–346. Zbl0543.14026
  15. [15] B. Saint-Donat, Projective models of K - 3 surfaces, Amer. J. Math.96 (1974), 602–639. Zbl0301.14011MR364263
  16. [16] P. Du Val, On surfaces whose canonical system is hyperelliptic, Canadian J. Math.4 (1952), 204–221. Zbl0046.39001MR48090
  17. [17] P. Griffiths and J. Harris, “Principles of Algebraic Geometry”, Wiley Classics Library, New York, 1994. Zbl0836.14001MR1288523
  18. [18] K. Konno, Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 575–595. Zbl0822.14009MR1267600
  19. [19] F. Polizzi, On surfaces of general type with p g = q = 1 , K 2 = 3 , Collect. Math.56 (2005), 181–234. Zbl1084.14038MR2154303
  20. [20] F. Polizzi, Surfaces of general type with p g = q = 1 , K 2 = 8 and bicanonical map of degree 2 , Trans. Amer. Math. Soc.358 (2006), 759–798. Zbl1083.14043MR2177040
  21. [21] M. Reid, Graded rings and birational geometry, In: “Proceedings of Algebraic Symposium” (Kinosaki, Oct 2000), K. Ohno (ed.) 1–72, available from www.maths. warwick.ac.uk/ miles/3folds 
  22. [22] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), 309–316. Zbl0663.14010MR932299
  23. [23] A. Todorov, A construction of surfaces with p g = 1 , q = 0 and 2 ≤ K 2 ≤ 8 . Counter examples of the global Torelli theorem, Invent. Math. 63 (1981), 287–304. Zbl0457.14016MR610540
  24. [24] G. Xiao, “Surfaces fibrées en courbes de genre deux”, Lecture Notes in Mathematics, Vol. 1137, Springer-Verlag, Berlin, 1985. Zbl0579.14028MR872271
  25. [25] G. Xiao, Degree of the bicanonical map of a surface of general type, Amer. J. Math.112 (1990), 713–736. Zbl0722.14025MR1073006

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.