The Moduli Space of (1,11)-Polarized Abelian Surfaces is Unirational

Compositio Mathematica

Volumn 126, Issue 1, 2001, Pages 1-24

  Gross, Mark ^a   Popescu, Sorin ^b,c  

^a UNIVERSITY OF WARWICK   (United Kingdom)

^b Och Spine at New York Presbyterian Hospitals   (United States)

^c STONY BROOK UNIVERSITY   (United States)

Author keywords

Abelian surfaces; Kleine's cubic hypersurface; Moduli

Indexed keywords

EID: 0042329597     PISSN: 0010437X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1017518027822     Document Type: Article

References (60)

1. Adler, A.: On the automorphism group of a certain cubic threefold, Amer. J. Math. 100(6) (1978), 1275-1280.
2. Adler, A.: Modular correspondences on X(11), Proc. Edinburgh Math. Soc. (2) 35(3) (1992), 427-435.
3. 5, Comm. Algebra 20(10) (1992), 2837-2862.
4. Adler, A.: Appendices to Moduli of Abelian Varieties, Lecture Notes in Math. 1644, Springer-Verlag, New York, 1996.
5. n, q = 2n ± 1, In: The Eightfold Way: The beauty of the Klein Quartic, MSRI Publ. 35, Cambridge Univ. Press, New York, 1999, pp. 175-219.
6. Adler, A. and Ramanan, S.: Moduli of Abelian Varieties, Lecture Notes in Math. 1644, Springer Verlag, New York, 1996.
7. Barth, W.: Quadratic equations for level-3 Abelian surfaces, In: Abelian Varieties (Egloffstein, 1993), de Gruyter, Berlin, 1995, pp. 1-18.
8. Barth, W. and Bauer, Th.: Smooth quartic surfaces with 352 conies, Manuscripta Math. 85(3-4) (1994), 409-417.
9. Bauer, Th.: Quartic surfaces with 16 skew conics, J. Reine Angew. Math. 464 (1995), 207-217.
10. Bayer, D. and Stillman, M.: Macaulay: A system for computation in algebraic geometry and commutative algebra Source and object code available for Unix and Macintosh computers. Contact the authors, or download from ftp://math.harvard.edu via anonymous ftp.
11. 4, In: Algebraic Threefolds (Varenna, 1981) Lecture Notes in Math. 947, Springer-Verlag, New York, 1982, pp. 190-208.
12. Behr, H. and Mennicke, J.: A presentation of the groups PSL(2, p), Canad. J. Math. 20 (1968), 1432-1438.
13. Borisov, L.: A finiteness theorem for subgroups of Sp(4, Z), preprint math.AG/9510002.
14. Buchsbaum, D. and Eisenbud, D.: Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Amer. J. Math. 99(3) (1977), 447-485.
15. Clemens, C. H. and Griffiths, P.: The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281-356.
16. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. With computational assistance from J. G. Thackray, Oxford University Press, Oxford, 1985.
17. Coble, A.: Point sets and allied Cremona groups III, Trans. Amer. Math. Soc. 18 (1917), 331-372.
18. Dolgachev, I.: Invariant stable bundles over modular curves X(p), Preprint math.AG/9710012.
19. Dornhoff, L.: Group Representation Theory, Marcel Dekker, New York, 1971/72.
20. Edge, W. L.: Klein's encounter with the simple group of order 660, Proc. London Math. Soc. (3) 24 (1972), 647-668.
21. Fano, G.: Nuove ricerche sulle varietà algebriche a tre dimensioni a curve-sezionicanoniche, Pontificia Acad. Sci. Coment. 11 (1947), 635-720.
22. Grant, D.: Formal groups in genus two, J. Reine Angew. Math. 411 (1990), 96-121.
23. Grayson, D. and Stillman, M.: Macaulay 2: A computer program designed to support computations in algebraic geometry and computer algebra. Source and object code available from http://www.math.uiuc.edu/Macaulay2/.
24. Gritsenko, V.: Irrationality of the moduli spaces of polarized Abelian surfaces. With an appendix by the author and K. Hulek, In: Abelian varieties (Egloffstein, 1993), 63-84, de Gruyter, Berlin, 1995.
25. Gritsenko, V.: Irrationality of the moduli spaces of polarized Abelian surfaces, Internat. Math. Res. Notices (1994), No. 6, 235 ff., approx. 9 pp. (electronic).
26. Gritsenko, V. and Hulek, K.: Commutator coverings of Siegel threefolds, Duke Math. J. 94(3) (1998), 509-542.
27. Gross, M. and Popescu, S.: Equations of (1, d)-polarized Abelian surfaces, Math. Ann. 310(2) (1998), 333-377.
28. Gross, M. and Popescu, S.: Calabi-Yau 3-folds and moduli of Abelian surfaces I, to appear in Compositio Math.
29. Gross, M. and Popescu, S.: Calabi-Yau 3-folds and moduli of Abelian surfaces II, In preparation.
30. Gushel, N. P.: Fano 3-folds of genus 8, Algebra i Analiz 4(1) (1992), 120-134.
31. 4 with 15,000 symmetries, Topology 12 (1973), 63-81.
32. Hulek, K.: Projective geometry of elliptic curves, Astérisque 137 1986.
33. Hulek, K., Kahn, C. and Weintraub, S.: Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, de Gruyter, Berlin, 1993.
34. Hulek, K. and Sankaran, G. K.: The Kodaira dimension of certain moduli spaces of Abelian surfaces, Compositio Math. 90(1) (1994), 1-35.
35. Hulek, K. and Sankaran, G. K.: The geometry of Siegel modular varieties, Preprint math.AG/9810153.
36. Iskovskih, V. A.: Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat. 42(3) (1978), 506-549.
37. Iskovskih, V. A.: Birational automorphisms of three-dimensional algebraic varieties, in Current Problems in Mathematics, Vol. 12, VINITI, Moscow, 1979, pp. 159-236, 239.
38. Klein, F.: Über die Transformation elfter Ordnung der elliptische Funktionen, Math. Ann. 15 (1879), [Ges. Math. Abh., Band III, art. LXXXVI, 140-168].
39. Klein, F.: Über die Transformation elfter Ordnung der elliptische Funktionen, Math. Ann. 15 (1879), [Ges. Math. Abh., Band III, art. LXXXVI, 140-168].
40. Klein, F. and Fricke, R.: Theorie der elliptischen Modulfunktionen Bd. I, Teubner, Leipzig 1890.
41. Lazarsfeld, R.: Projectivité normale des surfaces Abeliennes (written by O. Debarre), Preprint Europroj 14, Nice.
42. Lange, H. and Birkenhake, Ch.: Complex Abelian Varieties, Springer-Verlag, New York, 1992.
43. Manolache, N. and Schreyer, F.-O.: Moduli of (1,7)-polarized Abelian surfaces via syzygies, Preprint math.AG/9812121.
44. Mukai, S.: Curves, K3 surfaces and Fano 3-folds of genus ≤ 10, In: Algebraic Geometry and Commutative Algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 357-377.
45. Mumford, D.: Abelian Varieties, Oxford University Press, 1974.
46. Murre, J. P.: Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of Mumford, Compositio Math. 27 (1973), 63-82.
47. O'Grady, K. G.: On the Kodaira dimension of moduli spaces of Abelian surfaces, Compositio Math. 72(2) (1989), 121-163.
48. Puts, P.J.: On some Fano-threefolds that are sections of Grassmannians, Nederl. Akad. Wetensch. Indag. Math. 44(1) (1982), 77-90.
49. Revoy, Ph.: Formes alternées et puissances divisées, Séminaire Dubreil, 1972/1973.
50. Schreyer, F.-O.: Geometry and algebra of prime Fano 3-folds of index 1 and genus 12, Preprint 1997.
51. Sekiguchi, T.: On the cubics defining Abelian varieties, J. Math. Soc. Japan 30(4) (1978), 703-721.
52. Semple, G. and Roth, L.: Algebraic Geometry, Chelsea, 1937.
53. Silberger, A. J.: An elementary construction of the representations of SL(2, GF(q)), Osaka J. Math. 6 (1969), 329-338.
54. Takeuchi, K: Some birational maps of Fano 3-folds, Compositio Math. 71(3) (1989), 265-283.
55. Tanaka, S.: Construction and classification of irreducible representations of special linear group of the second order over a finite field, Osaka J. Math. 4 (1967), 65-84.
56. Tregub, S. L.: Construction of a birational isomorphism of a three-dimensional cubic and a Fano variety of the first kind with g = 8, connected with a normal rational curve of degree 4, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 113(6) (1985), 99-101. (English translation: Moscow Univ. Math. Bull. 40(6) (1985), 78-80.)
57. Tregub, S. L.: Construction of a birational isomorphism of a three-dimensional cubic and a Fano variety of the first kind with g = 8, connected with a normal rational curve of degree 4, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 113(6) (1985), 99-101. (English translation: Moscow Univ. Math. Bull. 40(6) (1985), 78-80.)
58. van der Geer, G.: On the geometry of a Siegel modular threefold, Math. Ann. 260(3) (1982), 317-350.
59. n, Bull. Soc. Math. France 57(5) (1978), 152.
60. Weil, A.: Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211.