The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products

2023/01/24 by Connelly, Erin, Agarwal, Sameer, Ergur, Alperen +1
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2301.09826

Abstract

Given k ≤ 6 points (x_i,y_i) ∈ ℙ² × ℙ², we characterize rank deficiency of the k × 9 matrix Z_k with rows x_i^\top ⊗ y_i^\top in terms of the geometry of the point configurations \x_i\ and \y_i\. While this question comes from computer vision the answer relies on tools from classical algebraic geometry: 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.

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