The geometry of rank drop in a class of face-splitting matrix products: Part I.

Given k points (xi, yi) ∈ ℙ2 × ℙ2, we characterize rank deficiency of the k × 9 matrix Zk with rows x i ⊤ ⊗ y i ⊤ in terms of the geometry of the point configurations {xi} and {yi}. In this paper we present results for k ≤ 6. For k ≤ 5, the geometry of the rank-drop locus is characterized by cross-ratios and basic (projective) geometry of point configurations. For the case k = 6 the rank-drop locus is captured by the classical theory of cubic surfaces. The results for k = 7, 8 and 9 are presented in the sequel [7] to this paper. [ABSTRACT FROM AUTHOR]