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

✍ Scribed by Laurent Busé; Jean-Pierre Jouanolou

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
376 KB
Volume
265
Category
Article
ISSN
0021-8693

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✦ Synopsis

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 ) ⊂ 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 n-2 k with k a field, and base points are either absent or local complete intersection isolated points.

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