The Library

Calabi–Yau threefolds and moduli of abelian surfaces I

Tools

Gross, M. W. (Mark W.), 1965- and Popescu, Sorin, 1963-. (2001) Calabi–Yau threefolds and moduli of abelian surfaces I. Compositio Mathematica, Vol.12 (No.2). pp. 169-228. ISSN 0010-437X

Preview

PDF
WRAP_Gross_Calabi_yau.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (511Kb)

Official URL: http://dx.doi.org/10.1023/A:1012076503121

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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Calabi-Yau manifolds, Abelian varieties, Moduli theory, Manifolds (Mathematics), Functions of several complex variables
Journal or Publication Title:	Compositio Mathematica
Publisher:	Cambridge University Press
ISSN:	0010-437X
Date:	June 2001
Volume:	Vol.12
Number:	No.2
Page Range:	pp. 169-228
Identification Number:	10.1023/A:1012076503121
Status:	Peer Reviewed
Access rights to Published version:	Open Access
Funder:	National Science Foundation (U.S.) (NSF), Mathematical Sciences Research Institute (Berkeley, Calif.) (MSRI)
Grant number:	DMS-9700761 (NSF), DMS-9610205 (NSF)
References:	[Ad] Adler, A.: On the automorphism group of a certain cubic threefold, Amer. J. Math. 100(6) (1978), 1275^1280. [Au] Aure, A.: Surfaces on quintic threefolds associated to the Horrocks^Mumford bundle, In: Lecture Notes in Math. 1399, Springer, New York, 1989, pp. 1^9. [ADHPR1] Aure, A. B., Decker, W., Hulek, K., Popescu, S. and Ranestad, K.: The geometry of bielliptic surfaces in P4, Internat. J. Math. 4 (1993), 873^902. [ADHPR2] Aure, A. B., Decker, W., Hulek, K., Popescu, S. and Ranestad, K.: Syzygies of Abelian and bielliptic surfaces in P4, Internat. J. Math. 8 (1997), 849^919. [Ba] Barth, W.: Abelian surfaces with (1,2) polarization, In: Algebraic Geometry (Sendai 1985), Adv. Stud. Pure Math. 10, Kinokuniya, Tokyo, 1985, pp. 41^84. [BM] Barth, W. andMoore R.: Geometry in the space of Horrocks^Mumford surfaces, Topology 28(2) (1989), 231^245. [BHM] Barth, W., Hulek, K., and Moore, R.: Degenerations of Horrocks^Mumford surfaces, Math. Ann. 277(4) (1987), 735^755. [BS] 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.columbia.edu via anonymous ftp. [Bea] Beauville, A.: Variëtës de Prym et jacobiennes intermëdiaires, Ann. Sci. École Norm. Sup. (4) 10(3) (1977), 309^391. [BeMe] Behr, H. and Mennicke, J.: A presentation of the groups PSL(2,p), Canad. J. Math. 20 (1968), 1432^1438. [BL] Birkenhake, Ch. and Lange, H.: Moduli spaces of Abelian surfaces with isogeny, In: Geometry and analysis (Bombay, 1992), Tata Inst. Fund. Res., Bombay, 1995, pp. 225^243. [BLvS] Birkenhake, Ch., Lange, H., and van Straten, D.: Abelian surfaces of type (1,4), Math. Ann. 285 (1989), 625^646. [Bor1] Borcea, C.: On desingularized Horrocks^Mumford quintics, J. Reine, Angew. Math. 421 (1991), 23^41. [Bor2] Borcea, C.: Nodal quintic threefolds and nodal octic surfaces, Proc. Amer. Math. Soc. 109(3) (1990), 627^635. [Bori] Borisov, L.: A ¢niteness theorem for Sp(4,Z), Algebraic geometry, 9, J. Math. Sci. (New York) 94(1), (1999), 1073^1099; math.AG/9510002. [Cle] Clemens, C. H.: Double solids, Adv. in Math. 47(3) (1983), 107^230. [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 Univ. Press, Oxford, 1985. [DHS] Debarre, O., Hulek, K., and Spandaw, J.: Very ample linear systems on abelian varieties, Math. Ann. 300 (1994), 181^202. [DK] Dolgachev, I. and Kanev, V.: Polar covariants of plane cubics and quartics, Adv. Math. 98(2) (1993), 216^301. [DO] Dolgachev, I. and Ortland, D.: Points sets in projective spaces and theta functions, Aste¨risque 165 (1988). [Do] Donagi, R.: The unirationality of A5 , Ann. of Math. (2) 119(2) (1984), 269^307. [EPW] Eisenbud, D., Popescu, S. and Walter, Ch.: Enriques surfaces and other non-Pfaffian subcanonical subschemes of codimension 3, Comm. Algebra 28(12) (2000), special volume for Robin Hartshorne's 60th birthday, to appear, math.AG/9906171. [Fry] Fryers, M.: The moveable cones of Calabi^Yau threefolds, Thesis, Univ. Cambridge 1998. [vG] van Geemen, B.: The moduli space of curves of genus 3 with level 2 structure is rational, Unpublished preprint. [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), de Gruyter, Berlin, 1995, pp. 63^84. [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). [GP1] Gross, M. and Popescu, S.: Equations of (1,d)-polarized Abelian surfaces, Math. Ann. 310 (1998), 333^377. [GP2] Gross,M. and Popescu, S.: The moduli space of (1,11)-polarized Abelian surfaces in unirational, Compositio Math. 126 (2001), 1^24. [GP3] Gross, M. and Popescu, S.: Calabi^Yau threefolds and moduli of Abelian surfaces II, in preparation. [Ha] Hartshorne, R.: Algebraic Geometry, Springer, New York, 1977. [Has] Hassett, B.: Some rational cubic fourfolds, J. Algebraic Geom. 8(1) (1999), 103^114. [HM] Horrocks, G. and Mumford, D.: A Rank 2 vector bundle on P4 with 15,000 symmetries, Topology, 12 (1973), 63^81. [Hu1] Hulek, K.: Projective geometry of elliptic curves, Astërisque 137 (1986). [Hu2] Hulek, K.: Geometry of the Horrocks^Mumford bundle, In: Algebraic Geometry (Bowdoin 1985), Proc. Symp. Pure Math. 46 (2), Amer. Math. Soc., Providence, p. 69^85. [HKW] Hulek, K., Kahn, C. and Weintraub, S.: Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions,Walter 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. [HS2] Hulek, K. and Sankaran, G. K.: The geometry of Siegel modular varieties, Preprint math.AG/9810153. [I] Igusa, J.: Arithmetic variety of moduli for genus two, Ann. ofMath. (2) 72 (1960), 612^649. [Isk] Iskovskikh, V. A.: Algebraic threefolds (a brief survey), (Russian) In: Algebra, Gos. Univ., Moscow, 1982, pp. 46^78. [Iy] Iyer, J. N.: Projective normality of Abelian surfaces given by primitive line bundles, Manuscripta Math. 98(2) (1999), 139^153. [Kat] Katsylo, P.: Rationality of the moduli variety of curves of genus 3, Comment. Math. Helv. 71(4) (1996), 507^524. [Kl1] Klein, F.: Ûber transformationen siebenter Ordnung der elliptischen Funktionen (1878/79), Abhandlung LXXXIV, in Gesammelte Werke, Bd. III, Springer, Berlin, 1924. [Kl2] Klein, F.: Über die elliptischen Normalkurven der n-ten Ordnung (1885), Abhandlung XC, in Gesammelte Werke, Bd. III, Springer, Berlin, 1924. [La] Lazarsfeld, R.: Projectivitë normale des surfaces Abeliennes, (written by O. Debarre) Pre-print Europroj 14, Nice. [LB] Lange, H. and Birkenhake, Ch.: Complex Abelian Varieties, Springer-Verlag, New York, 1992. [MS] Manolache, N. and Schreyer, F.-O.: Moduli of (1,7)-polarized Abelian surfaces via syzygies, Math. Nachr. (2001) to appear, preprint math.AG/9812121. [Mie] Miele, M.: Klassifikation der Durchscnitte Heisenberg-invarianter Systeme von Quadriken in P6, Thesis, Erlangen, 1995. [Mi] Miyata, T.: Invariants of certain groups I, Nagoya Math. J. 41 (1971), 69^73. [Moo] Moore, R.: Heisenberg-invariant quintic 3-folds, and sections of the Horrocks^Mumford bundle, Preprint, Canberra 1985. [MM] Mori, S. and Mukai, S.: The uniruledness of the moduli space of curves of genus 11, In: Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer, New York, 1983, pp. 334^353. [Muk1] Mukai, S.: Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Nat. Acad. Sci. U.S.A. 86(9) (1989), 3000^3002. [Muk2] Mukai, S.: Fano 3-folds, In: Complex Projective Geometry (Trieste, 1989/Bergen, 1989), LondonMath. Soc. Lecture Note Ser. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 255. [Mu1] Mumford, D.: On the equations de¢ning Abelian varieties, Invent. Math. 1 (1966), 287^354. [Mu2] Mumford, D.: Abelian Varieties, Oxford Univ. Press, 1974. [Mu3] Mumford, D.: Tata Lectures on Theta I, Progress in Math. 28, Birkhäuser, Basel, 1983. [O'G] O'Grady, K. G., On the Kodaira dimension of moduli spaces of Abelian surfaces, Compositio Math. 72(2) (1989), 121^163. [Og] Oguiso, K., On algebraic ¢ber space structures on a Calabi^Yau 3-fold, With an appendix by Noboru Nakayama. Internat. J. Math. 4(3) (1993), 439^465. [RS] Ranestad, K. and Schreyer, F.-O.: Varieties of sums of powers, J. Reine Angew. Math. 525 (2000), 147^181. [Rod] Rodland, E.: The Pffafian Calabi^Yau, its mirror, and their link to the Grassmannian G(2,7), Compositio Math. 122(2) (2000), 135^149. [Sal] Salmon, G.: Modern Higher Algebra 4, Hodges, Figgis, and Co., Dublin, 1885. [Scho] Schoen, C.: Algebraic cycles on certain desingularized nodal hypersurfaces, Math. Ann. 270 (1985), 17^27. [Scho2] Schoen, C.: On the geometry of a special determinantal hypersurface associated to the Horrocks^Mumford vector bundle, J. Reine Angew. Math. 364 (1986), 85^111. [Schr] Schreyer, F.-O.: Geometry and algebra of prime Fano 3-folds of index 1 and genus 12, Compositio Math. (2001), to appear, preprint math.AG/9911044. [SR] Semple, G. and Roth, L.: Algebraic Geometry, Chelsea, 1937. [Si] Silberger, A. J.: An elementary construction of the representations of SL(2,GF(q)), Osaka J. Math. 6 (1969), 329^338. [Ta] 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. [Tei] Teissier, B.: The hunting of invariants in the geometry of discriminants, In: Real and Complex Singularities, (Proc.Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 565^678. [Tjo] Tjotta, E.: Quantum cohomology of a Pfaffian Calabi^Yau variety: verifying mirror symmetry predictions, Compositio Math. 126 (2001), 79^89. [Ve] Vëlu, J.: Courbes elliptiques munies d'un sous-groupe Z=nZ mn , Bull. Soc. Math. France, Me¨m. 57 (1978), 5^152. [Ver] Verra, A.: A short proof of the unirationality of A5, Nederl. Akad. Wetensch. Indag. Math. 46(3) (1984), 339^355. [We] Werner, J.: Kleine Aufläsungen spezieller dreidimensionaler Varietäten, Bonner Math. Schriften 186, Universität Bonn, Mathematisches Institut, Bonn, 1987.
URI:	http://wrap.warwick.ac.uk/id/eprint/818