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Wahl maps and extensions of canonical curves and K ⁢ 3 K3 surfaces

Authors
  • Ciliberto, Ciro1
  • Dedieu, Thomas2
  • Sernesi, Edoardo3
  • 1 Università degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 , (Italy)
  • 2 Université de Toulouse CNRS, UPS IMT, 31062, Cedex 9 , (France)
  • 3 Università Roma Tre, Largo S.L. Murialdo 1, 00146 , (Italy)
Type
Published Article
Journal
Journal für die reine und angewandte Mathematik (Crelles Journal)
Publisher
De Gruyter
Publication Date
Jul 11, 2018
Volume
2020
Issue
761
Pages
219–245
Identifiers
DOI: 10.1515/crelle-2018-0016
Source
De Gruyter
License
Yellow

Abstract

Let C be a smooth projective curve (resp. ( S , L ) {(S,L)} a polarized K ⁢ 3 {K3} surface) of genus g ⩾ 11 {g\geqslant 11} , with Clifford index at least 3, considered in its canonical embedding in ℙ g - 1 {\mathbb{P}^{g-1}} (resp. in its embedding in | L | ∨ ≅ ℙ g {|L|^{\vee}\cong\mathbb{P}^{g}} ). We prove that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in ℙ g + r {\mathbb{P}^{g+r}} , not a cone, with dim ⁡ ( Y ) = r + 2 {\dim(Y)=r+2} and ω Y = 𝒪 Y ⁢ ( - r ) {\omega_{Y}=\mathcal{O}_{Y}(-r)} , if the cokernel of the Gauss–Wahl map of C (resp. H 1 ⁡ ( T S ⊗ L ∨ ) {\operatorname{H}^{1}(T_{S}\otimes L^{\vee})} ) has dimension larger than or equal to r + 1 {r+1} (resp. r). This relies on previous work of Wahl and Arbarello–Bruno–Sernesi. We provide various applications.

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