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

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

2008-03-05

arxiv.org/abs/0803.0573v2

Mathematics

Algebraic Geometry

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

Scientific paper

We develop in this paper some methods for studying the implicitization problem for a rational map $\phi: \mathbb{P}^n \to (\mathbb{P}^1)^{n+1}$ 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 $\phi$, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants.

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