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Let  C  be a smooth projective curve (resp.(  S  ,  L  ){(S,L)}    a polarizedK  ⁢  3{K3}    surface) of genusg  ⩾  11{g\geqslant 11}   , with Clifford index at least 3, considered in its canonical embedding inℙ   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  inℙ   g  +  r{\mathbb{P}^{g+r}}   , not a cone, withdim  ⁡   (  Y  )    =   r  +  2{\dim(Y)=r+2}    andω  Y   =      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 tor  +  1{r+1}    (resp.  r ). This relies on previous work of Wahl and Arbarello–Bruno–Sernesi. We provide various applications.

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