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

TL;DR

This paper introduces methods for solving the implicitization problem of a rational map from projective space to a product of projective lines, using Koszul complexes and approximation complexes, with applications to sparse discriminants.

Contribution

It develops a unified approach linking Macaulay Resultants and Koszul complexes for implicitization, providing geometric interpretations and conditions for scheme-theoretic images.

Findings

Methods for computing implicit equations using Koszul complexes.

Conditions for the implicit equation to define the scheme-theoretic image.

Applications to computing sparse discriminants.

Abstract

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.