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Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds

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In this article we exhibit certain projective degenerations of smoothK3 surfaces of degree 2g−2 in ℙg (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of suchK3 surfaces has a corank one Gaussian map, ifg=11 org≥13. We also prove that the general such hyperplane section lies on a uniqueK3 surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which does not rely on the existence of a line.

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Authors and Affiliations

  1. Dipartimento di Matematica, Universita di Roma II, Via Fontanile di Carcaricola, 00173, Roma, Italy

    Ciro Ciliberto

  2. Dipartimento di Matematica, Universita di Pavia, Strada Nuova 65, 27100, Pavia, Italy

    Angelo Lopez

  3. Department of Mathematics, Colorado State University, 80523, Fort Collins, CO, USA

    Rick Miranda

Authors
  1. Ciro Ciliberto
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  2. Angelo Lopez
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  3. Rick Miranda
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Oblatum 1-II-1993 & 24-V-1993

Research supported in part by NSF grant DMS-9104058

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Ciliberto, C., Lopez, A. & Miranda, R. Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds. Invent Math 114, 641–667 (1993). https://doi.org/10.1007/BF01232682

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