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

Abstract

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.