The implicitization problem for $\varphi :P^n-->(P^1{)}^{n+1}$

Abstract

We develop in this paper some methods for studying the implicitization problem for a rational map $\varphi :{\mathbb{P}}^n\to ({\mathbb{P}}^1{)}^{n+1}$ defining a hypersurface in $({\mathbb{P}}^1{)}^{n+1}$, 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, up to a power, and up to some other extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of $\varphi$, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants.

17 pages, revised version. To appear in Journal of Algebra