Resumen de Non-surjective Gaussian maps for singular curves on K3 surfaces

Claudio Fontanari, Edoardo Sernesi

• Let (S, L) be a polarized K3 surface with \mathrm {Pic}(S) = \mathbb {Z}[L] and L\cdot L=2g-2, let C be a nonsingular curve of genus g-1 and let f:C\rightarrow S be such that f(C) \in \vert L \vert. We prove that the Gaussian map \Phi _{\omega _C(-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 \widetilde{S} of S at x and a theorem of L’vovski.