Computing the Chow variety of quadratic space curves

TL;DR

This paper investigates the structure of the Chow variety of quadratic space curves, decomposing it into specific subvarieties and computing their defining ideals within a high-dimensional projective space.

Contribution

It explicitly computes the ideals of subvarieties of coisotropic hypersurfaces related to quadratic space curves, extending the understanding of their algebraic structure.

Findings

Decomposition of the Chow variety into Chow forms of conics and pairs of lines, and Hurwitz forms of quadric surfaces.

Explicit computation of the ideals defining these subvarieties.

Enhanced understanding of the algebraic and geometric structure of quadratic space curves.

Abstract

Quadrics in the Grassmannian of lines in 3-space form a 19-dimensional projective space. We study the subvariety of coisotropic hypersurfaces. Following Gel'fand, Kapranov and Zelevinsky, it decomposes into Chow forms of plane conics, Chow forms of pairs of lines, and Hurwitz forms of quadric surfaces. We compute the ideals of these loci.