On the closed image of a rational map and the implicitization problem

In this paper, we investigate some topics around the closed image S of a rational map λ given by some homogeneous elements f1,…,fn of the same degree in a graded algebra A. We first compute the degree of this closed image in case λ is generically finite and f1,…,fn define isolated base points in Proj(A). We then relate the definition ideal of S to the symmetric and the Rees algebras of the ideal I=(f1,…,fn) A, and prove some new acyclicity criteria for the associated approximation complexes. Finally, we use these results to obtain the implicit equation of S in case S is a hypersurface, with k a field, and base points are either absent or local complete intersection isolated points.