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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:
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