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 criterions 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 with k a field, and base points are either absent or local complete intersection isolated points.