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 $f_1,...,f_n$ 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 $f_1,...,f_n$ 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 $I=(f_1,...,f_n) \subset A$, 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, $\Proj(A)=\PP^{n-2}_k$ with $k$ a field, and base points are either absent or local complete intersection isolated points.
43 pages, revised version. To appear in Journal of Algebra
43 pages, revised version. To appear in Journal of Algebra