  In this paper the author provides a generalization of classical linkage, i.e.
linkage by a complete intersection of dim. 0 or 1 on arithmetically
Cohen-Macaulay schemes of any dimension. Namely she looks at residuals in the
scheme theoretic intersection of an aCM scheme of dim. r (resp. r+1) with r
hypersurfaces of degree a_1,...,a_r. When the aCM scheme is singular a complete
intersection on it may not be Gorenstein, in this case classical linkage, even
if suitably generalized, does not apply. The main purpose of this paper is to
prove some results related to the invariance of the deficiency module under
such linkage. In the last part of the paper the author shows how to apply these
results and techniques to the classification of curves C in P^n of degree d and
maximal genus G(d, n, s) among those not contained in surfaces of degree less
than a certain fixed one s.
