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

Laurent Busé 1, 2 Jean-Pierre Jouanolou 3
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : In this paper, we investigate some topics around the closed image $S$ of a rational map $\lambda$ given by some homogeneous elements $f_1,\ldots,f_n$ 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 $f_1,\ldots,f_n$ 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 $I=(f_1,\ldots,f_n) \subset A$, 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.
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Laurent Busé, Jean-Pierre Jouanolou. On the closed image of a rational map and the implicitization problem. Journal of Algebra, Elsevier, 2003, 265 (1), pp.312--357. ⟨inria-00098682⟩

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