The implicitization problem for φ{symbol} : Pn (P1)n + 1
We develop in this paper methods for studying the implicitization problem for a rational map φ{symbol} : Pn (P1)n + 1 defining a hypersurface in (P1)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 co...
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2009
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v322_n11_p3878_Botbol http://hdl.handle.net/20.500.12110/paper_00218693_v322_n11_p3878_Botbol |
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| Sumario: | We develop in this paper methods for studying the implicitization problem for a rational map φ{symbol} : Pn (P1)n + 1 defining a hypersurface in (P1)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, up to a power, and up to some extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of φ{symbol}, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants. © 2009 Elsevier Inc. All rights reserved. |
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