  We study the surjectivity of suitable weighted Gaussian maps which provide a
natural generalization of the standard Gaussian maps and encode the local
geometry of the locus of curves endowed with a higher root of the canonical
bundle having associated linear system of dimension at least r + 1. In
particular, we get a bound on the dimension of its Zariski tangent space, which
turns out to be sharp in the special case r = 0. Finally, we describe this
locus in the case of complete intersection curves.
