Lines, Quadrics, and Cremona Transformations in Two-View Geometry

Given $7\le k\le 9$ points $(x_i,y_i)\in {\mathbb{P}}^2\times {\mathbb{P}}^2$, we characterize rank deficiency of the $k\times 9$ matrix $Z_k$ with rows $x_i^{\mathrm{\top }}\otimes y_i^{\mathrm{\top }}$, in terms of the geometry of the point sets $\{x_i\}$ and $\{y_i\}$. This problem arises in the conditioning of certain well-known reconstruction algorithms in computer vision, but has surprising connections to classical algebraic geometry via the interplay of quadric surfaces, cubic curves and Cremona transformations. The characterization of rank deficiency of $Z_k$, when $k\le 6$, was completed in arXiv:2301.09826.