Classification of quadruple Galois canonical covers I
HTML articles powered by AMS MathViewer
- by Francisco Javier Gallego and Bangere P. Purnaprajna PDF
- Trans. Amer. Math. Soc. 360 (2008), 5489-5507 Request permission
Abstract:
In this article we classify quadruple Galois canonical covers of smooth surfaces of minimal degree. The classification shows that they are either non-simple cyclic covers or bi-double covers. If they are bi-double, then they are all fiber products of double covers. We construct examples to show that all the possibilities in the classification do exist. There are implications of this classification that include the existence of families with unbounded geometric genus, in sharp contrast with triple canonical covers, and families with unbounded irregularity, in sharp contrast with canonical covers of all other degrees. Together with the earlier known results on double and triple covers, a pattern emerges that motivates some general questions on the existence of higher degree canonical covers, some of which are answered in this article.References
- Lucian Bădescu, Algebraic surfaces, Universitext, Springer-Verlag, New York, 2001. Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author. MR 1805816, DOI 10.1007/978-1-4757-3512-3
- Arnaud Beauville, L’application canonique pour les surfaces de type général, Invent. Math. 55 (1979), no. 2, 121–140 (French). MR 553705, DOI 10.1007/BF01390086
- Mauro C. Beltrametti and Tomasz Szemberg, On higher order embeddings of Calabi-Yau threefolds, Arch. Math. (Basel) 74 (2000), no. 3, 221–225. MR 1739501, DOI 10.1007/s000130050434
- F. Catanese, On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984), no. 2, 483–515. MR 755236
- Francisco Javier Gallego and B. P. Purnaprajna, Very ampleness and higher syzygies for Calabi-Yau threefolds, Math. Ann. 312 (1998), no. 1, 133–149. MR 1645954, DOI 10.1007/s002080050215
- Francisco Javier Gallego and Bangere P. Purnaprajna, On the canonical rings of covers of surfaces of minimal degree, Trans. Amer. Math. Soc. 355 (2003), no. 7, 2715–2732. MR 1975396, DOI 10.1090/S0002-9947-03-03200-8
- Francisco Javier Gallego and Bangere P. Purnaprajna, Classification of quadruple canonical covers: Galois case, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), no. 2, 45–50 (English, with French summary). MR 2055225
- Francisco Javier Gallego and Bangere P. Purnaprajna, Classification of quadruple Galois canonical covers. II, J. Algebra 312 (2007), no. 2, 798–828. MR 2333185, DOI 10.1016/j.jalgebra.2006.11.011
- Mark L. Green, The canonical ring of a variety of general type, Duke Math. J. 49 (1982), no. 4, 1087–1113. MR 683012
- David W. Hahn and Rick Miranda, Quadruple covers of algebraic varieties, J. Algebraic Geom. 8 (1999), no. 1, 1–30. MR 1658196
- Eiji Horikawa, Algebraic surfaces of general type with small $C^{2}_{1}.$ I, Ann. of Math. (2) 104 (1976), no. 2, 357–387. MR 424831, DOI 10.2307/1971050
- Kazuhiro Konno, Algebraic surfaces of general type with $c^2_1=3p_g-6$, Math. Ann. 290 (1991), no. 1, 77–107. MR 1107664, DOI 10.1007/BF01459239
- Madhav V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 305–344. MR 732347
- Keiji Oguiso and Thomas Peternell, On polarized canonical Calabi-Yau threefolds, Math. Ann. 301 (1995), no. 2, 237–248. MR 1314586, DOI 10.1007/BF01446628
- Ulf Persson, Double coverings and surfaces of general type, Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977) Lecture Notes in Math., vol. 687, Springer, Berlin, 1978, pp. 168–195. MR 527234
Additional Information
- Francisco Javier Gallego
- Affiliation: Departamento de Álgebra, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Email: gallego@mat.ucm.es
- Bangere P. Purnaprajna
- Affiliation: Department of Mathematics, 405 Snow Hall, University of Kansas, Lawrence, Kansas 66045-2142
- Email: purna@math.ku.edu
- Received by editor(s): November 15, 2006
- Published electronically: May 28, 2008
- Additional Notes: The first author was partially supported by MCT project number BFM2000-0621. He is grateful for the hospitality of the Department of Mathematics of the University of Kansas at Lawrence.
The second author is grateful to NSA and the GRF of the University of Kansas for supporting this research project. He is also grateful for the hospitality of the Departamento de Álgebra of the Universidad Complutense de Madrid - © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 5489-5507
- MSC (2000): Primary 14J10, 14J26, 14J29
- DOI: https://doi.org/10.1090/S0002-9947-08-04587-X
- MathSciNet review: 2415082
Dedicated: Dedicated to Ignacio Sols