  Here we investigate the canonical Gaussian map for higher multiple coverings
of curves, the case of double coverings being completely understood thanks to
previous work by Duflot. In particular, we prove that every smooth curve can be
covered with degree not too high by a smooth curve having arbitrarily large
genus and surjective canonical Gaussian map. As a consequence, we recover an
asymptotic version of a recent result by Ciliberto and Lopez and we address the
Arbarello subvariety in the moduli space of curves.
