Non-surjective Gaussian maps for singular curves on K3 surfaces

Abstract

Let (S, L) be a polarized K3 surface with Pic(?)=ℤ[?] and ?⋅?=2?−2 , let C be a nonsingular curve of genus ?−1 and let ?:?→? be such that ?(?)∈|?| . We prove that the Gaussian map Φ??(−?) 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 ?˜ of S at x and a theorem of L’vovski.