Higher Gaussian Maps on K3 surfaces

We give necessary conditions for the surjectivity of the higher Gaussian maps on a polarized K3 surface. As an application, we show that the higher $k$-th Gauss map for a general curve of genus $g$ (that depends quadratically with $k$) is surjective.
Comment: 20 pages; v2: Final version. Accepted for publication in IMRN