## Nagoya Mathematical Journal

### On the second Gaussian map for curves on a K3 surface

#### Abstract

By a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first Gaussian map is not surjective. In this paper we prove that if $C$ is a general hyperplane section of high genus (> 280) of a general polarized K3 surface, then the second Gaussian map of $C$ is surjective. The resulting bound for the genus $g$ of a general curve with surjective second Gaussian map is decreased to $g$.

#### Article information

Source
Nagoya Math. J., Volume 199 (2010), 123-136.

Dates
First available in Project Euclid: 14 September 2010

https://projecteuclid.org/euclid.nmj/1284471573

Digital Object Identifier
doi:10.1215/00277630-2010-006

Mathematical Reviews number (MathSciNet)
MR2730414

Zentralblatt MATH identifier
1208.14021

#### Citation

Colombo, Elisabetta; Frediani, Paola. On the second Gaussian map for curves on a K3 surface. Nagoya Math. J. 199 (2010), 123--136. doi:10.1215/00277630-2010-006. https://projecteuclid.org/euclid.nmj/1284471573

#### References

• [1] E. Ballico and C. Fontanari, On the surjectivity of higher Gaussian maps for complete intersection curves, Ricerche Mat. 53 (2004), 79–85.
• [2] A. Beauville, Preliminaires sur les periodes des surfaces K3, Asterisque 126 (1985), 91–97.
• [3] A. Beauville and J.-Y. Merindol, Sections hyperplanes des surfaces K3, Duke Math. J. 55 (1987), 873–878.
• [4] C. Ciliberto, J. Harris, and R. Miranda, On the surjectivity of the Wahl map, Duke Math. J. 57 (1988), 829–858.
• [5] C. Ciliberto, A. F. Lopez, and R. Miranda, Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds, Invent. Math. 114 (1993), 641–667.
• [6] C. Ciliberto, A. F. Lopez, and R. Miranda, “On the corank of Gaussian maps for general embedded K3 surfaces” in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc. 9, Bar-Ilan University, Ramat Gan, 1996, 141–157.
• [7] C. Ciliberto, A. F. Lopez, and R. Miranda, Classification of varieties with canonical curve section via Gaussian maps on canonical curves, Amer. J. Math. 120 (1998), 1–21.
• [8] E. Colombo and P. Frediani, Some results on the second Gaussian map for curves, Michigan Math. J. 58 (2009), 745–758.
• [9] E. Colombo and P. Frediani, Siegel metric and curvature of the moduli space of curves, Trans. Am. Math. Soc. 362 (2010), no. 3, 1231–1246.
• [10] E. Colombo, P. Frediani, and G. Pareschi, Hyperplane sections of abelian surfaces, preprint, to appear in J. Algebraic Geom., arXiv:math/0903.2781
• [11] E. Colombo, G. P. Pirola, and A. Tortora, Hodge-Gaussian maps, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 30 (2001), 125–146.
• [12] M. L. Green, “Infinitesimal methods in Hodge theory” in Algebraic Cycles and Hodge Theory, Torino 1993, Lect. Notes Math. 1594, Springer, Berlin, 1994, 1–92.
• [13] Y. Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann. 261 (1982), 43–46.
• [14] S. Mori, On degrees and genera of curves on smooth quartic surfaces in P3, Nagoya Math. J. 96 (1984), 127–132.
• [15] D. R. Morrison, On K3 surfaces with large Picard number, Invent. Math. 75 (1984), 105–121.
• [16] B. Saint-Donat, Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602–639.
• [17] E. Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1–8.
• [18] C. Voisin, Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math. 168 (1992), 249–272.
• [19] J. Wahl, The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J. 55 (1987), 843–871.
• [20] J. Wahl, Gaussian maps on algebraic curves, J. Differential Geom. 32 (1990), 77–98.
• [21] J. Wahl, “Introduction to Gaussian maps on an algebraic curve” in Complex Projective Geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lect. Note Ser. 179, Cambridge University Press, Cambridge, 1992, 304–323.