Hyperplane sections of abelian surfaces
Authors:
Elisabetta Colombo, Paola Frediani and Giuseppe Pareschi
Journal:
J. Algebraic Geom. 21 (2012), 183-200
DOI:
https://doi.org/10.1090/S1056-3911-2011-00556-0
Published electronically:
April 25, 2011
MathSciNet review:
2846682
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Abstract |
References |
Additional Information
Abstract: By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of abelian surfaces. The somewhat surprising result is that the Wahl map of such curves is (tendentially) surjective, but their second Wahl map has corank at least 2 (in fact a more precise result is proved).
References
- Edoardo Ballico and Claudio Fontanari, On the surjectivity of higher Gaussian maps for complete intersection curves, Ricerche Mat. 53 (2004), no. 1, 79–85 (2005). MR 2290553
- A. Beauville and J.-Y. Mérindol, Sections hyperplanes des surfaces $K3$, Duke Math. J. 55 (1987), no. 4, 873–878 (French). MR 916124, DOI https://doi.org/10.1215/S0012-7094-87-05541-4
- Ciro Ciliberto, Joe Harris, and Rick Miranda, On the surjectivity of the Wahl map, Duke Math. J. 57 (1988), no. 3, 829–858. MR 975124, DOI https://doi.org/10.1215/S0012-7094-88-05737-7
- Elisabetta Colombo and Paola Frediani, Some results on the second G aussian map for curves, Michigan Math. J. 58 (2009), no. 3, 745–758. MR 2595562, DOI https://doi.org/10.1307/mmj/1260475698
- Elisabetta Colombo and Paola Frediani, Siegel metric and curvature of the moduli space of curves, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1231–1246. MR 2563728, DOI https://doi.org/10.1090/S0002-9947-09-04845-4
- Elisabetta Colombo and Paola Frediani, On the second Gaussian map for curves on a $K3$ surface, Nagoya Math. J. 199 (2010), 123–136. MR 2730414, DOI https://doi.org/10.1215/00277630-2010-006
- Elisabetta Colombo, Gian Pietro Pirola, and Alfonso Tortora, Hodge-Gaussian maps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 1, 125–146. MR 1882027
- Jaya N. Iyer, Projective normality of abelian surfaces given by primitive line bundles, Manuscripta Math. 98 (1999), no. 2, 139–153. MR 1667600, DOI https://doi.org/10.1007/s002290050131
- Herbert Lange and Christina Birkenhake, Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. MR 1217487
- Robert Lazarsfeld, Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), no. 3, 299–307. MR 852158
- Lazarsfeld, R., Projectivité normale des surfaces abeliennes, Redigé par O. Debarre, Prépub 14, C.I.M.P.A., Nice (1990).
- Shigefumi Mori and Shigeru Mukai, The uniruledness of the moduli space of curves of genus $11$, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 334–353. MR 726433, DOI https://doi.org/10.1007/BFb0099970
- Giuseppe Pareschi, A proof of Lazarsfeld’s theorem on curves on $K3$ surfaces, J. Algebraic Geom. 4 (1995), no. 1, 195–200. MR 1299009
- Giuseppe Pareschi, Gaussian maps and multiplication maps on certain projective varieties, Compositio Math. 98 (1995), no. 3, 219–268. MR 1351829
- Paris, M.: La proprietà di Petri per curve su superfici abeliane, Doctorate Thesis, Università di Roma, La Sapienza, 2000 (unpublished).
- Claire Voisin, Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math. 168 (1992), no. 3-4, 249–272 (French). MR 1161267, DOI https://doi.org/10.1007/BF02392980
- Jonathan M. Wahl, The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J. 55 (1987), no. 4, 843–871. MR 916123, DOI https://doi.org/10.1215/S0012-7094-87-05540-2
- Jonathan Wahl, Introduction to Gaussian maps on an algebraic curve, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 304–323. MR 1201392, DOI https://doi.org/10.1017/CBO9780511662652.023
- Jonathan Wahl, On cohomology of the square of an ideal sheaf, J. Algebraic Geom. 6 (1997), no. 3, 481–511. MR 1487224
- Calabri, A., Ciliberto, C., Miranda, R. The rank of the $2$nd Gaussian map for general curves, arXiv:0911.4734
References
- Ballico, E., Fontanari, C., On the surjectivity of higher Gaussian maps for complete intersection curves. Ricerche Mat. 53 (2004), no. 1, 79–85 (2005). MR 2290553 (2007k:14096)
- Beauville, A., Mérindol, J.-Y., Sections hyperplanes des surfaces K3, Duke Math. Jour. 55, no. 4, (1987), 873–878. MR 916124 (89a:14043)
- Ciliberto, C., Harris, J., Miranda, R. On the surjectivity of the Wahl map, Duke Math. Jour. 57, (1988), 829–858. MR 975124 (89m:14010)
- Colombo, E., Frediani, P., Some results on the second Gaussian map for curves, Michigan Math. J. 58 (2009), no. 3, 745–758. MR 2595562 (2011c:14095)
- Colombo, E., Frediani, P., Siegel metric and curvature of the moduli space of curves, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1231–1246. MR 2563728 (2010h:14046)
- Colombo, E., Frediani, P., On the second Gaussian map for curves on K3 surfaces, Nagoya Math. J. 199 (2010), 123–136. MR 2730414
- Colombo, E., Pirola, G.P., Tortora, A., Hodge-Gaussian maps, Ann. Scuola Normale Sup. Pisa Cl. Sci. (4) 30 (2001), no. 1, 125–146. MR 1882027 (2002k:32034)
- Iyer, J., Projective normality of abelian surfaces given by primitive line bundles, Manuscripta Mat. 98 (1999), 139-153. MR 1667600 (2000b:14056)
- Birkenhake, Ch., Lange, H., Complex Abelian Varieties, 2nd edition, Grundlehren der Mathematischen Wissenschaften, 302. Springer-Verlag, Berlin, 2000. MR 1217487 (94j:14001)
- Lazarsfeld, R., Brill-Noether-Petri without degeneration, J. of Differential Geom. (3) 23 (1986), 299-307. MR 852158 (88b:14019)
- Lazarsfeld, R., Projectivité normale des surfaces abeliennes, Redigé par O. Debarre, Prépub 14, C.I.M.P.A., Nice (1990).
- Mori, S., Mukai, S., The uniruledness of the moduli space of curves of genus 11, in Algebraic Geometry Proceedings Tokyo, Kyoto, 1982, Springer LNM 1016 (1983), 334–353. MR 726433 (85b:14033)
- Pareschi, G., A proof of Lazarsfeld’s theorem on curves on K3 surfaces, J. of Alg. Geom. 4 (1995), 195–200. MR 1299009 (95m:14022)
- Pareschi, G., Gaussian maps and multiplication maps on certain projective varieties. Compositio Math. 98 (1995), no. 3, 219–268. MR 1351829 (97c:14027)
- Paris, M.: La proprietà di Petri per curve su superfici abeliane, Doctorate Thesis, Università di Roma, La Sapienza, 2000 (unpublished).
- Voisin, C., Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math. 168 (1992), 249–272. MR 1161267 (93b:14045)
- Wahl, J., The Jacobian algebra of a graded Gorenstein singularity, Duke Math. Jour. 55 (1987), 843–871. MR 916123 (89a:14042)
- Wahl, J., Introduction to Gaussian maps on an algebraic curve, Complex projective geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge (1992), 304–323. MR 1201392 (93m:14029)
- Wahl, J., On the cohomology of the square of an ideal sheaf, J. of Alg. Geom. 8 (1997), 481–512. MR 1487224 (99b:14046)
- Calabri, A., Ciliberto, C., Miranda, R. The rank of the $2$nd Gaussian map for general curves, arXiv:0911.4734
Additional Information
Elisabetta Colombo
Affiliation:
Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133, Milano, Italy
Email:
elisabetta.colombo@unimi.it
Paola Frediani
Affiliation:
Dipartimento di Matematica, Università di Pavia, via Ferrata 1, I-27100 Pavia, Italy
MR Author ID:
347739
ORCID:
0000-0003-2537-2727
Email:
paola.frediani@unipv.it
Giuseppe Pareschi
Affiliation:
Dipartimento di Matematica, Università di Roma, Tor Vergata, V.le della Ricerca Scientifica, I-00133 Roma, Italy
Email:
pareschi@mat.uniroma2.it
Received by editor(s):
May 7, 2009
Received by editor(s) in revised form:
November 9, 2009
Published electronically:
April 25, 2011
Additional Notes:
Partially supported by PRIN 2007 MIUR: “Spazi dei moduli e teoria di Lie” and PRIN 2006 of MIUR “Geometry on algebraic varieties”.
