Fitting ideals and multiple points of surface parameterizations

Given a birational parameterization ϕ   of an algebraic surface S⊂P3S⊂P3, the purpose of this paper is to investigate the sets of points on SS whose pre-image consists of k or more points, counting multiplicities. These points are described explicitly in terms of Fitting ideals of some graded parts of the symmetric algebra associated with the parameterization ϕ. To obtain this description, we show that the degree and dimension of a fiber could be computed by comparing the drop of rank of two explicit (representation) matrices associated with ϕ.