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On the second Gaussian map for curves on a $$K3$$ surface. (English) Zbl 1208.14021
For a curve $$X$$, the higher Gaussian maps $$\mu_i$$ of the canonical bundle $$K_X$$ are higher order generalisations of the classical Gauss map of the canonical embedding of $$X$$ (see for example [J. M. Wahl, Lond. Math. Soc. Lect. Note Ser. 179, 304–323 (1992; Zbl 0790.14014)]). It is known that the first Gauss map $$\mu_1$$ is surjective for a general curve of genus $$g\geq 10$$, $$g\neq 11$$ [see C. Ciliberto, J. Harris and R. Miranda, Duke Math. J. 57, No. 3, 829–858 (1988; Zbl 0684.14009); C. Voisin, Acta Math. 168, No. 3–4, 249–272 (1992; Zbl 0767.14012)] and that it cannot be surjective for curves whose canonical embedding lies on a $$K3$$ surface [J. M. Wahl, Duke Math. J. 55, 843–871 (1987; Zbl 0644.14001)].
The paper under review deals with similar questions, but in the case of the second Gaussian map. Specifically, the main result is that the second Gaussian map is surjective for curves of high genus ($$>280$$) that are general hyperplane sections of a general polarized $$K3$$ surface. This yields that $$\mu_2$$ is surjective for general curves of genus $$g>125$$. Although these results are not expected to be optimal (surjectivity can be expected for general curves of genus $$\geq18$$), this construction gives the first known lower bound.
Other examples in which the surjectivity of the second Gaussian map is known are curves lying on products of two curves [E. Colombo and P. Frediani, Mich. Math. J. 58, No. 3, 745–758 (2009; Zbl 1191.14030)] and complete intersections [E. Ballico and C. Fontanari, Ric. Mat. 53, No. 1, 79–85 (2004; Zbl 1221.14035)].

MSC:
 14H10 Families, moduli of curves (algebraic) 14J28 $$K3$$ surfaces and Enriques surfaces
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References:
 [1] E. Ballico and C. Fontanari, On the surjectivity of higher Gaussian maps for complete intersection curves , Ricerche Mat. 53 (2004), 79-85. · Zbl 1221.14035 [2] A. Beauville, Preliminaires sur les periodes des surfaces K 3, Asterisque 126 (1985), 91-97. · Zbl 0577.14028 [3] A. Beauville and J.-Y. Merindol, Sections hyperplanes des surfaces K3 , Duke Math. J. 55 (1987), 873-878. · Zbl 0663.14028 [4] C. Ciliberto, J. Harris, and R. Miranda, O n the surjectivity of the Wahl map , Duke Math. J. 57 (1988), 829-858. · Zbl 0684.14009 [5] C. Ciliberto, A. F. Lopez, and R. Miranda, Projective degenerations of K 3 surfaces, Gaussian maps, and Fano threefolds , Invent. Math. 114 (1993), 641-667. · Zbl 0807.14028 [6] C. Ciliberto, A. F. Lopez, and R. Miranda, “On the corank of Gaussian maps for general embedded K 3 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. · Zbl 0865.14019 [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. · Zbl 0934.14028 [8] E. Colombo and P. Frediani, Some results on the second Gaussian map for curves , Michigan Math. J. 58 (2009), 745-758. · Zbl 1191.14030 [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. · Zbl 1196.14025 [10] E. Colombo, P. Frediani, and G. Pareschi, Hyperplane sections of abelian surfaces , preprint, to appear in J. Algebraic Geom., · Zbl 1281.14021 [11] E. Colombo, G. P. Pirola, and A. Tortora, Hodge-Gaussian maps , Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 30 (2001), 125-146. · Zbl 1018.14001 [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. · Zbl 0846.14001 [13] Y. Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theorem , Math. Ann. 261 (1982), 43-46. · Zbl 0476.14007 [14] S. Mori, On degrees and genera of curves on smooth quartic surfaces in P 3 , Nagoya Math. J. 96 (1984), 127-132. · Zbl 0576.14032 [15] D. R. Morrison, On K 3 surfaces with large Picard number , Invent. Math. 75 (1984), 105-121. · Zbl 0509.14034 [16] B. Saint-Donat, Projective models of K 3 surfaces , Amer. J. Math. 96 (1974), 602-639. JSTOR: · Zbl 0301.14011 [17] E. Viehweg, Vanishing theorems , J. Reine Angew. Math. 335 (1982), 1-8. · Zbl 0485.32019 [18] C. Voisin, Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri , Acta Math. 168 (1992), 249-272. · Zbl 0767.14012 [19] J. Wahl, The Jacobian algebra of a graded Gorenstein singularity , Duke Math. J. 55 (1987), 843-871. · Zbl 0644.14001 [20] J. Wahl, Gaussian maps on algebraic curves , J. Differential Geom. 32 (1990), 77-98. · Zbl 0724.14022 [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. · Zbl 0790.14014
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