The Segre Determinant

TL;DR

This paper investigates the Segre determinant, a polynomial encoding points on bilinear hypersurfaces, exploring its computation, geometric interpretation, and applications in algebraic vision and Grassmannian quotients.

Contribution

It provides new insights into the Segre determinant, linking it to Chow-Lam forms and demonstrating its applications in algebraic geometry and vision.

Findings

Segre determinant encodes bilinear hypersurface conditions.

It represents the Chow-Lam form of a generic torus orbit in Grassmannians.

Applications include algebraic vision and Chow quotients of Grassmannians.

Abstract

The Segre determinant is a polynomial which encodes the condition for points to lie on a bilinear hypersurface in the product of projective spaces. We study Segre determinants and compute them in various coordinate systems. We show that the Segre determinant represents the Chow-Lam form of a generic torus orbit in the Grassmannian. These Chow-Lam forms were introduced as a generalization of Chow forms for projective varieties, and enjoy many similar properties. We also present applications to algebraic vision and to Chow quotients of Grassmannians.