Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Gross, M. W.; Popescu, Sorin (2001)
Publisher: Cambridge University Press
Languages: English
Types: Article
Subjects: QA

Classified by OpenAIRE into

arxiv: Mathematics::Algebraic Geometry, Quantitative Biology::Cell Behavior, Physics::History of Physics
We prove that the moduli space $\cal A$11lev of <$>(1,11)-polarized Abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface in P4. Therefore, $\cal A$11lev is unirational but not rational, and there are no Γ11-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of $\cal A$9lev.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • Adler, A.: Appendices to Moduli of Abelian Varieties, Lecture Notes in Math. 1644, Springer-Verlag, New York, 1996.
    • Press, New York, 1999, pp. 175^219.
    • 1644, Springer Verlag, New York, 1996.
    • Barth, W.: Quadratic equations for level-3 Abelian surfaces, In: Abelian Varieties (Egloffstein, 1993), de Gruyter, Berlin, 1995, pp. 1^18.
    • Barth, W. and Bauer, Th.: Smooth quartic surfaces with 352 conics, Manuscripta Math. 85(3^4) (1994), 409^417.
    • Bauer, Th.: Quartic surfaces with 16 skew conics, J. Reine Angew. Math. 464 (1995), 207^217.
    • [Bea] Beauville, A.: Les singularit e¨s du diviseur Y de la jacobienne interm e¨diaire de l'hypersurface cubique dans P4, In: Algebraic Threefolds (Varenna, 1981) Lecture Notes in Math. 947, Springer-Verlag, New York, 1982, pp. 190^208.
    • [BE] Buchsbaum, D. and Eisenbud, D.: Algebra structures for ¢nite free resolutions, and some structure theorems for ideals of codimension 3. Amer. J. Math. 99(3) (1977), 447^485.
    • [CG] Clemens, C. H. and Grif¢ths, P.: The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281^356.
    • [CNPW] 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.
    • [Co] Coble, A.: Point sets and allied Cremona groups III, Trans. Amer. Math. Soc. 18 (1917), 331^372.
    • [Dor] Dornhoff, L.: Group Representation Theory, Marcel Dekker, New York, 1971/72.
    • [Ed] Edge, W. L.: Klein's encounter with the simple group of order 660, Proc. London Math. Soc. (3) 24 (1972), 647^668.
    • [Fa] Fano, G.: Nuove ricerche sulle varieta© algebriche a tre dimensioni a curve-sezionicanoniche, Ponti¢cia Acad. Sci. Coment. 11 (1947), 635-720.
    • [Gra] Grant, D.: Formal groups in genus two, J. Reine Angew. Math. 411 (1990), 96^121.
    • [GS] 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/.
    • [Gri1] 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.
    • [Gri2] Gritsenko, V.: Irrationality of the moduli spaces of polarized Abelian surfaces, Internat. Math. Res. Notices (1994), No. 6, 235 ff., approx. 9 pp. (electronic).
    • [GH] Gritsenko, V. and Hulek, K.: Commutator coverings of Siegel threefolds, Duke Math. J. 94(3) (1998), 509^542.
    • [GP2] Gross, M. and Popescu, S.: Calabi^Yau 3-folds and moduli of Abelian surfaces I, to appear in Compositio Math.
    • [GP3] Gross, M. and Popescu, S.: Calabi^Yau 3-folds and moduli of Abelian surfaces II, In preparation.
    • [Gu] Gushel, N. P.: Fano 3-folds of genus 8, Algebra i Analiz 4(1) (1992), 120^134.
    • [HM] Horrocks, G. and Mumford, D.: A Rank 2 vector bundle on P4 with 15,000 symmetries, Topology 12 (1973), 63^81.
    • [Hul] Hulek, K.: Projective geometry of elliptic curves, Aste¨ risque 137 1986.
    • [HKW] Hulek, K., Kahn, C. and Weintraub, S.: Moduli Spaces of Abelian Surfaces: Compacti¢cation, Degenerations, and Theta Functions, de Gruyter, Berlin, 1993.
    • [HS1] Hulek, K. and Sankaran, G. K.: The Kodaira dimension of certain moduli spaces of Abelian surfaces, Compositio Math. 90(1) (1994), 1^35.
    • Hulek, K. and Sankaran, G. K.: The geometry of Siegel modular varieties, Preprint math.AG/9810153.
    • Iskovskih, V. A.: Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat. 42(3) (1978), 506^549.
    • Iskovskih, V. A.: Birational automorphisms of three-dimensional algebraic varieties, in Current Problems in Mathematics, Vol. 12, VINITI, Moscow, 1979, pp. 159^236, 239.
    • Klein, F.: Uë ber die Transformation elfter Ordnung der elliptische Funktionen, Math. Ann. 15 (1879), [Ges. Math. Abh., Band III, art. LXXXVI, 140^168].
    • Klein, F. and Fricke, R.: Theorie der elliptischen Modulfunktionen Bd. I, Teubner, Leipzig 1890.
    • Debarre), Preprint Europroj 14, Nice.
    • Lange, H. and Birkenhake, Ch.: Complex Abelian Varieties, Springer-Verlag, New York, 1992.
    • Manolache, N. and Schreyer, F.-O.: Moduli of (1,7)-polarized Abelian surfaces via syzygies, Preprint math.AG/9812121.
    • Mukai, S.: Curves, K3 surfaces and Fano 3-folds of genus W 10, In: Algebraic Geometry and Commutative Algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp.
    • Mumford, D.: Abelian Varieties, Oxford University Press, 1974.
    • 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.
    • O'Grady, K. G.: On the Kodaira dimension of moduli spaces of Abelian surfaces, Compositio Math. 72(2) (1989), 121^163.
    • Akad. Wetensch. Indag. Math. 44(1) (1982), 77^90.
    • Revoy, Ph.: Formes altern e¨es et puissances divise¨ es, Se¨ minaire Dubreil, 1972/1973.
    • Schreyer, F.-O.: Geometry and algebra of prime Fano 3-folds of index 1 and genus 12, Preprint 1997.
    • Sekiguchi, T.: On the cubics de¢ning Abelian varieties, J. Math. Soc. Japan 30(4) (1978), 703^721.
    • Semple, G. and Roth, L.: Algebraic Geometry, Chelsea, 1937.
    • Takeuchi, K: Some birational maps of Fano 3-folds, Compositio Math. 71(3) (1989), 265^283.
    • Tanaka, S.: Construction and classi¢cation of irreducible representations of special linear group of the second order over a ¢nite ¢eld, Osaka J. Math. 4 (1967), 65^84.
    • Tregub, S. L.: Construction of a birational isomorphism of a three-dimensional cubic and a Fano variety of the ¢rst kind with g ˆ 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.) van der Geer, G.: On the geometry of a Siegel modular threefold, Math. Ann. 260(3) (1982), 317^350.
    • France 57(5) (1978), 152.
    • Weil, A.: Sur certains groupes d'ope¨ rateurs unitaires, Acta Math. 111 (1964), 143^211.
  • No related research data.
  • No similar publications.

Share - Bookmark

Funded by projects

  • NSF | Equations and Moduli of Abe...
  • NSF | Calabi-Yau Threefolds and B...

Cite this article