  The BRST anomaly which may be present in the induced $W_n$ gravity quantized
on the light-cone is evaluated in the geometrical framework of Zucchini. The
cocycles linked by the cohomology of the BRST operator to the anomaly are
straightforwardly calculated thanks to the analogy between this formulation and
the Yang-Mills theory. We give also a conformally covariant formulation of
these quantities including the anomaly, which is valid on arbitrary Riemann
surfaces. The example of the $W_3$ theory is discussed and a comparison with
other candidates for the anomaly available in the literature is presented.
