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

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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 (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, (A)=ⁿ⁻²ₖ with k a field, and base points are either absent or local complete intersection isolated points.