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

Let C be a smooth projective curve (resp. {(S,L)} a polarized {K3} surface) of genus {g\geqslant 11}, with Clifford index at least 3, considered in its canonical embedding in {\mathbb{P}^{g-1}} (resp. in its embedding in {|L|^{\vee}\cong\mathbb{P}^{g}}). We prove that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in {\mathbb{P}^{g+r}}, not a cone, with {\dim(Y)=r+2} and {\omega_{Y}=\mathcal{O}_{Y}(-r)}, if the cokernel of the Gauss–Wahl map of C (resp. {\operatorname{H}^{1}(T_{S}\otimes L^{\vee})}) 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.

On the number of moduli of extendable canonical curves

Let C ⊂ ℙg−1 be a canonical curve of genus g. In this article we study the problem of extendability of C, that is when there is a surface S ⊂ ℙg different from a cone and having C as hyperplane section. Using the work of Epema we give a bound on the number of moduli of extendable canonical curves. This for example implies that a family of large dimension of curves that are cover of another curve has general member nonextendable. Using a theorem of Wahl we prove the surjectivity of the Wahl map for the general k-gonal curve of genus g when k = 5, g ≥ 15 or k = 6, g ≥ 13 or k ≥ 7, g ≥ 12.