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Surjectivity of Gaussian maps for curves on Enriques surfaces. (English) Zbl 1124.14035

Let \(C\) be a smooth projective algebraic curve and \(L,M\) line bundles on \(C\). Denote the canonical bundle of \(C\) by \(\omega_C\), and the kernel of the multiplication map \(H^0(L)\otimes H^0(M)\rightarrow H^0(L\otimes M)\) by \(R(L,M)\). The Gaussian map \(\Phi_{L,M}: R(L,M)\rightarrow H^0(C,\omega_C\otimes L\otimes M)\) is the map defined locally by \(\Phi_{L,M}(s\otimes t)=sdt-tds\) (for an introduction to Gaussian maps, see e.g. [J. Wahl, Lond. Math. Soc. Lect. Note Ser. 179, 304–323 (1992; Zbl 0790.14014)]). The subject of this paper is the study of \(\Phi_{L,M}\) in the special case where \(M\) is the canonical bundle of \(C\). Specifically, sufficients conditions are given, ensuring the surjectivity of \(\Phi_{L,\omega_C}\) for curves \(C\) lying on an Enriques surface. This result is used by the authors and R. Muñoz [On the extendability of projective surfaces and a genus bound for Enriques-Fano threefolds, preprint arXiv:math/0605750] to prove that the sectional genus of threefolds whose general hyperplane section is an Enriques surface cannot exceed \(17\).
The proof of the main result is obtained by combining suitable generalizations of known results about Gaussian maps and a specific study of curves on Enriques surfaces. In this latter stage, the authors show that the canonical model of a tetragonal curve of genus \(g\) lying on an Enriques surface and general in its linear system cannot be a quadric section of a surface of degree \(g-1\) if \(g\geq 7\).

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14H51 Special divisors on curves (gonality, Brill-Noether theory)

Citations:

Zbl 0790.14014
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References:

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