The implicitization problem for φ : Pn → (P1)n+l

JOURNAL OF ALGEBRA

卷:322

Article

Botbol, Nicolas^1,2 

[1] Univ Buenos Aires, Dept Matemat, FCEN, RA-1053 Buenos Aires, DF, Argentina

[2] Univ Paris 06, Inst Math Jussieu, F-75252 Paris 05, France

关键词: Elimination theory;    Rational map;    Syzygy;    Approximation complex;    Koszul complex;    Implicitization;   

DOI  :  10.1016/j.jalgebra.2009.03.006

来源: Elsevier

【 摘 要 】

We develop in this Paper methods for Studying the implicitization problem for a rational map phi : P-n -> (P-1)(n+1) defining a hypersurface in (P-1)(n+1), based On Computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay resultants and Koszul complexes coincides. in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions. Under suitable hypotheses. these techniques enable us to obtain the implicit equation, Lip to a power, and Lip to some extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of phi, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free