On surfaces which dominate Kummer surfaces
Abstract: We study isogeny relations between surfaces and Kummer surfaces. Specifically, we prove a Torelli-type theorem for the existence of rational maps from surfaces to Kummer surfaces, and a Kummer sandwich theorem for surfaces with Shioda-Inose structure.
- 1. Chen, X. Self rational maps of surfaces. Preprint, arXiv:1008.1619.
- 2. Hiroshi Inose, Defining equations of singular 𝐾3 surfaces and a notion of isogeny, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 495–502. MR 578868
- 3. Shouhei Ma, On the 0-dimensional cusps of the Kähler moduli of a 𝐾3 surface, Math. Ann. 348 (2010), no. 1, 57–80. MR 2657434, 10.1007/s00208-009-0466-x
- 4. Afsaneh Mehran, Double covers of Kummer surfaces, Manuscripta Math. 123 (2007), no. 2, 205–235. MR 2306633, 10.1007/s00229-007-0092-4
- 5. D. R. Morrison, On 𝐾3 surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105–121. MR 728142, 10.1007/BF01403093
- 6. V. V. Nikulin, Finite groups of automorphisms of Kählerian 𝐾3 surfaces, Trudy Moskov. Mat. Obshch. 38 (1979), 75–137 (Russian). MR 544937
- 7. V. V. Nikulin, On rational maps between 𝐾3 surfaces, Constantin Carathéodory: an international tribute, Vol. I, II, World Sci. Publ., Teaneck, NJ, 1991, pp. 964–995. MR 1130874
- 8. Tetsuji Shioda, Kummer sandwich theorem of certain elliptic 𝐾3 surfaces, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 137–140. MR 2279280
- 9. T. Shioda and H. Inose, On singular 𝐾3 surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136. MR 0441982
- Chen, X. Self rational maps of surfaces. Preprint, arXiv:1008.1619.
- Inose, H. Defining equations of singular surfaces and a notion of isogeny. Proceedings of the International Symposium on Algebraic Geometry, pp. 495-502, Kinokuniya Book Store, 1978. MR 578868 (81h:14021)
- Ma, S. On the 0-dimensional cusps of the Kähler moduli of a surface. Math. Ann. 348 (2010), no. 1, 57-80. MR 2657434 (2011g:14094)
- Mehran, A. Double cover of Kummer surfaces. Manuscripta Math. 123 (2007), 205-235. MR 2306633 (2008c:14052)
- Morrison, D. R. On surfaces with large Picard number. Invent. Math. 75 (1984), no. 1, 105-121. MR 728142 (85j:14071)
- Nikulin, V. V. Finite groups of automorphisms of Kahlerian surfaces. Trudy Moskov. Mat. Obshch. 38 (1979), 75-137. MR 544937 (81e:32033)
- Nikulin, V. V. On rational maps between surfaces. Constantin Carathéodory: An international tribute, 964-995, World Sci. Publ., 1991. MR 1130874 (92m:14049)
- Shioda, T. Kummer sandwich theorem of certain elliptic surfaces. Proc. Japan Acad. Ser. A. 82 (2006), no. 8, 137-140. MR 2279280 (2008b:14064)
- Shioda, T.; Inose, H. On singular surfaces. Complex analysis and algebraic geometry, pp. 119-136. Iwanami Shoten, 1977. MR 0441982 (56:371)
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
Address at time of publication: Graduate School of Mathematics, Nagoya University, Furō-chō, Chikusa-ku, Nagoya 464-8602, Japan
Email: firstname.lastname@example.org, email@example.com
Keywords: $K3$ surface, rational map, Shioda-Inose structure.
Received by editor(s): March 17, 2011
Received by editor(s) in revised form: June 12, 2011
Published electronically: May 15, 2012
Communicated by: Lev Borisov
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.