Given a birational parameterization f of an algebraic surface S, the purpose of this paper is toinvestigate the sets of points on S whose preimage consists in kor more points, counting multiplicities. These points are described explicitly interms of Fitting ideals of some graded parts of the symmetric algebraassociated to the parameterization f. To obtain this description, weshow that the degree and dimension of a fiber could be computed bycomparing the drop of rank of two explicit (representation) matrices associated to f.