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

Given k points (x(i), y(i)) is an element of P-2 x P-2, we characterize rank deficiency of the k x 9 matrix Z(k) with rows x(i)(inverted perpendicular)circle times y(i)(inverted perpendicular) in terms of the geometry of the point configurations {x(i)} and {y(i)}. 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.