Non-surjective Gaussian maps for singular curves on K3 surfaces

Let (S,L) be a polarized K3 surface with Pic(S)=Z[L] and L⋅L=2g−2, let C be a nonsingular curve of genus g−1 and let f:C→S be such that f(C)∈|L|. We prove that the Gaussian map ΦωC(−T) is non-surjective, where T is the degree two divisor over the singular point x of f(C). This generalizes a result of Kemeny with an entirely different proof. It uses the very ampleness of C on the blown-up surface S˜ of S at x and a theorem of L'vovski. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM.