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Wahl maps and extensions of canonical curves and K3 surfaces

  • Ciro Ciliberto , Thomas Dedieu and Edoardo Sernesi

Abstract

Let C be a smooth projective curve (resp. (S,L) a polarized K3 surface) of genus g11, with Clifford index at least 3, considered in its canonical embedding in g-1 (resp. in its embedding in |L|g). We prove that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in g+r, not a cone, with dim(Y)=r+2 and ωY=𝒪Y(-r), if the cokernel of the Gauss–Wahl map of C (resp. H1(TSL)) has dimension larger than or equal to r+1 (resp. r). This relies on previous work of Wahl and Arbarello–Bruno–Sernesi. We provide various applications.

Award Identifier / Grant number: 652782

Funding statement: Ciro Ciliberto and Thomas Dedieu were members of the project FOSICAV, which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 652782.

Acknowledgements

We thank (in alphabetical order) Gavin Brown, Cinzia Casagrande, Andreas Höring, Andreas Knutsen, and Serge Lvovski, for their kind and inspiring answers to our questions. We also thank the anonymous referee for valuable comments and suggestions.

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Received: 2018-03-16
Revised: 2018-05-07
Published Online: 2018-07-11
Published in Print: 2020-04-01

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