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Heisenberg-invariant kummer surfaces

Published online by Cambridge University Press:  20 January 2009

K. Hulek ,
I. Nieto  and
G. K. Sankaran
K. Hulek
Affiliation:
Institut für Mathematik, Universität Hannover, Postfach 6009, D 30060 Hannover, Germany ([email protected])
I. Nieto
Affiliation:
Cimat, A.C., Callejon de Jalisco S/N, Col. Mineral de Valenciana, 36000 Guanajuato, Gto., Mexico ([email protected])
G. K. Sankaran
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK ([email protected])
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Abstract

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We study, from the point of view of abelian and Kummer surfaces and their moduli, the special quintic threefold known as Nieto's quintic. It is, in some ways, analogous to the Segre cubic and the Burkhardt quartic and can be interpreted as a moduli space of certain Kummer surfaces. It contains 30 planes and has 10 singular points: we describe how some of these arise from bielliptic and product abelian surfaces and their Kummer surfaces.

Keywords

abelian surfaceKummer surfacemoduliclassical algebraic geometry
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 2 , June 2000 , pp. 425 - 439
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1. Barth, W. and Nieto, I., Abelian surfaces of type (1,3) and quartic surfaces with 16 skew lines, J. Alg. Geom. 3 (1994), 173–222.Google Scholar
2. Bauer, T., Projective images of Kummer surfaces, Math. Ann. 299 (1994), 155–170.CrossRefGoogle Scholar
3. Gritsenko, V. and Hulek, K., Minimal Siegel modular threefolds, Math. Proc. Camb. Phil. Soc. 123 (1998), 461–485.CrossRefGoogle Scholar
4. Hulek, K. and Weintraub, S., Bielliptic abelian surfaces, Math. Ann. 283 (1989), 411–429.CrossRefGoogle Scholar
5. Hulek, K., Nieto, I. and Sankaran, G. K., Degenerations of abelian surfaces of type (1,3) and Kummer surfaces, in Algebraic Geometry: Hirzebruch 70 (ed. Pragacz, P., Szurek, M. and Wiśniewski, J.) (AMS Contemporary Mathematics, 1999).Google Scholar
6. Hudson, R. W. H. T., Kummer's quartic surface (Cambridge University Press, 1906; reissued 1990).Google Scholar
7. Hunt, B., The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, no. 1637 (Springer, Berlin, 1996).CrossRefGoogle Scholar
8. Jessop, C. M., Quartic surfaces with singular points (Cambridge University Press, 1916).Google Scholar
9. Lange, H. and Birkenhake, Ch., Complex abelian varieties (Springer, Berlin, 1992).CrossRefGoogle Scholar
10. Mumford, D., On the equations defining abelian varieties, I, Inv. Math. 1 (1966), 287–354.CrossRefGoogle Scholar
11. Naruki, I., On smooth quartic embedding of Kummer surfaces, Proc. Jap. Acad. A 67 (1991), 223–225.Google Scholar
12. Nieto, I., Invariante Quartiken unter der Heisenberg Gruppe T, PhD thesis, University of Erlangen, 1989.Google Scholar
13. Nieto, I., Examples of abelian surfaces with polarization type (1,3), in Algebraic geometry and singularities (ed. Lopez, C. and Macarro, N.), Progress in Mathematics, vol. 134, pp. 319–337 (Birkhauser, Basel, 1996).CrossRefGoogle Scholar
14. Nikulin, V., On Kummer surfaces, Math. USSR. Izv. 9 (1975), 261–275.CrossRefGoogle Scholar