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Calabi–Yau threefolds and moduli of abelian surfaces I

By M. W. Gross and Sorin Popescu

Abstract

We describe birational models and decide the rationality/unirationality of moduli spaces $\cal A$d (and $\cal A$levd) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero

Topics: QA
Publisher: Cambridge University Press
Year: 2001
OAI identifier: oai:wrap.warwick.ac.uk:818

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Citations

  1. 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/.
  2. (1999). A ¢niteness theorem for Sp(4,Z), Algebraic geometry,
  3. (1968). A presentation of the groups PSL(2,p), doi
  4. (1973). A Rank 2 vector bundle on P4 with 15,000 symmetries, doi
  5. (1984). A short proof of the unirationality of A5, doi
  6. 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.columbia.edu via anonymous ftp.
  7. (1989). Abelian surfaces of type (1,4), doi
  8. (1985). Abelian surfaces with (1,2) polarization, In: Algebraic Geometry doi
  9. (1974). Abelian Varieties, doi
  10. (1937). Algebraic Geometry, doi
  11. (1982). Algebraic threefolds (a brief survey),
  12. (1969). An elementary construction of the representations of SL(2,GF(q)),
  13. (1960). Arithmetic variety of moduli forgenustwo, doi
  14. (1989). Biregularclassi¢cation of Fano 3-foldsand Fano manifoldsof coindex 3,
  15. (1985). C o n w a y ,J .H . ,C u r t i s ,R .T . ,N o r t o n ,S .P . ,P a r k e r ,R .A .a n dW i l s o n ,R .A . : Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups, with computational assistance from
  16. Calabi^Yau threefolds and moduli of Abelian surfaces II, in preparation. doi
  17. (1992). Ch.: Complex Abelian Varieties, doi
  18. (1967). Construction and classi¢cation of irreducible representations of special linear group of the second order over a ¢nite ¢eld,
  19. (1978). Courbes elliptiques munies d'un sous-groupe Z=nZ mn,
  20. (1987). Degenerations of Horrocks^Mumford surfaces, doi
  21. (1983). Double solids, doi
  22. (2000). Enriques surfaces and other non-Pfaf¢an subcanonical subschemes of codimension 3, doi
  23. (1998). Equations of (1,d)-polarized Abelian surfaces, doi
  24. (1989). Fano 3-folds, In: Complex Projective Geometry (Trieste, doi
  25. (2001). Geometry and algebra of prime Fano 3-folds of index 1 and genus 12, Compositio Math.
  26. (1989). Geometry in the space of Horrocks^Mumford surfaces, doi
  27. (1985). Geometry of the Horrocks^Mumford bundle, In: AlgebraicGeometry (Bowdoin doi
  28. (1985). Heisenberg-invariant quintic 3-folds, and sections of the Horrocks^ Mumford bundle, Preprint,
  29. (1971). Invariants of certain groups I,
  30. (1994). Irrationality of the moduli spaces of polarized Abelian surfaces, doi
  31. (1993). Irrationality of the moduli spaces of polarized Abelian surfaces. With an appendix by the author and K. Hulek, In: AbelianVarieties (Egloffstein, doi
  32. (1995). Klassi¢kation der Durchscnitte Heisenberg-invarianter Systeme von Quadriken in P6, Thesis,
  33. (1987). Kleine Au£Îsungen spezieller dreidimensionaler VarietÌten, Bonner Math. Schriften 186,
  34. (1885). Modern Higher Algebra 4, Hodges, Figgis, and Co.,
  35. (2001). Moduli of (1,7)-polarized Abelian surfaces via syzygies, doi
  36. (1995). Moduli spaces of Abelian surfaces with isogeny, In: Geometryandanalysis (Bombay,1992),
  37. (1993). Moduli Spaces of Abelian Surfaces: Compacti¢cation,Degenerations,andThetaFunctions, Walter de Gruyter, doi
  38. (1993). On algebraic ¢ber space structures on a Calabi^Yau 3-fold, With an appendix by Noboru Nakayama. doi
  39. (1991). On desingularized Horrocks^Mumford quintics, doi
  40. (1978). On the automorphism group of a certain cubic threefold, doi
  41. (1966). On the equations de¢ning Abelian varieties, doi
  42. (1986). On the geometry of a special determinantal hypersurface associated to the Horrocks^Mumford vector bundle, doi
  43. (1989). On the Kodaira dimension of moduli spaces of Abelian surfaces, doi
  44. (1988). Points sets in projective spaces and theta functions,
  45. (1993). Polar covariants of plane cubics and quartics, doi
  46. (1986). Projective geometry of elliptic curves, doi
  47. (1999). Projective normality of Abelian surfaces given by primitive line bundles, doi
  48. Projectivite ¨ normale des surfaces Abeliennes, (written by O.
  49. (2001). Quantum cohomology of a Pfaf¢an Calabi^Yau variety: verifying mirror symmetry predictions,
  50. (1996). Rationality of the moduli variety of curves of genus 3, doi
  51. (1999). Some rational cubic fourfolds,
  52. (1989). Surfaces on quintic threefolds associated to the Horrocks^Mumford bundle, In: doi
  53. (1997). Syzygies of Abelian and bielliptic surfaces in P4, doi
  54. (1993). The geometry of bielliptic surfaces in P4, doi
  55. The geometry of Siegel modular varieties, doi
  56. (1976). The hunting of invariants in the geometry of discriminants, In:Real andComplexSingularities,(Proc.NinthNordicSummerSchool/NAVFSympos. doi
  57. (2001). The moduli space of (1,11)-polarized Abelian surfaces in unirational,
  58. The moduli space of curves of genus 3 with level 2 structure is rational, doi
  59. (1998). The moveable cones of Calabi^Yau threefolds, Thesis, Univ.
  60. (2000). The Pffa¢an Calabi^Yau, its mirror, and their link to the Grassmannian G(2,7),
  61. (1984). The unirationality of A5, doi
  62. (1982). The uniruledness of the moduli space of curves of genus 11, In: Algebraic Geometry (Tokyo/Kyoto, doi
  63. (1924). Ûber die elliptischen Normalkurven der n-ten Ordnung (1885), Abhandlung XC, doi
  64. (1924). Ûber transformationen siebenter Ordnung der elliptischen Funktionen (1878/79), Abhandlung LXXXIV, doi
  65. (1977). Varie ¨ te ¨ s de Prym et jacobiennes interme ¨ diaires,
  66. (2000). Varieties of sums of powers, doi
  67. (1994). Very ample linear systems on abelian varieties, doi

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