Transcendental lattice of an extremal elliptic surface
Author:
Alex Degtyarev
Journal:
J. Algebraic Geom. 21 (2012), 413-444
DOI:
https://doi.org/10.1090/S1056-3911-2011-00563-8
Published electronically:
June 27, 2011
MathSciNet review:
2914799
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Abstract |
References |
Additional Information
Abstract: We develop an algorithm computing the transcendental lattice and the Mordell–Weil group of an extremal elliptic surface. As an example, we compute the lattices of four exponentially large series of surfaces.
References
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References
- K. Arima, I. Shimada, Zariski–van Kampen method and transcendental lattices of certain singular $K3$ surfaces, Tokyo J. Math. 32, no. 1 (2009), 201–227. MR 2541164 (2010i:14066)
- F. Beukers, H. Montanus, Explicit calculation of elliptic fibrations of $K3$-surfaces and their Belyi-maps, Number theory and polynomials, London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, Cambridge, (2008), 33–51. MR 2428514 (2009j:14011)
- A. Degtyarev, Zariski $k$-plets via dessins d’enfants, Comment. Math. Helv. 84, no. 3, (2009), 639–671. MR 2507257 (2010f:14028)
- R. Friedman, J. W. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer–Verlag, Berlin–New York, (1994). MR 1288304 (95m:57046)
- E. R. van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255–260. MR 1506962
- K. Kodaira, On compact analytic surfaces, II–III, Annals of Math., 77–78 (1963), 563–626, 1–40. MR 0184257 (32:1730)
- V. V. Nikulin, Integer quadratic forms and some of their geometrical applications, Izv. Akad. Nauk SSSR, Ser. Mat 43 (1979), 111–177, Russian, English transl. in Math. USSR–Izv. 14 (1980), 103–167. MR 525944 (80j:10031)
- M. Nori, On certain elliptic surfaces with maximal Picard number, Topology 24 (1985), no. 2, 175–186. MR 793183 (86m:14022)
- T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 0429918 (55:2927)
- T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul 39 (1990), no. 2, 211–240. MR 1081832 (91m:14056)
Additional Information
Alex Degtyarev
Affiliation:
Department of Mathematics, Bilkent University, 06800 Ankara, Turkey
MR Author ID:
271103
ORCID:
0000-0001-6586-4094
Email:
degt@fen.bilkent.edu.tr
Received by editor(s):
July 10, 2009
Received by editor(s) in revised form:
September 29, 2009, and December 31, 2009
Published electronically:
June 27, 2011
