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. given by some homogeneous elements f(1),..., f(n) of the same degree in a graded algebra A. We first compute the degree of this closed image in case lambda is generically finite and f(1),..., f(n) 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 = (f(1),..., f(n)) c 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, Proj(A) = P-k(n-2) with k a field, and base points are either absent or local complete intersection isolated points. (C) 2003 Elsevier Science (USA). All rights reserved.