Beyond the Kummer Surface: Quotients of Quadratic Line Complexes and Certified Focal Geometry
I-AIMS preprint sh-185 · 2026
A quadratic line complex is the intersection of the Klein quadric of lines in projective three-space with a second quadric in Plücker space. Its focal, or singular, surface is generically a Kummer quartic. The Kummer surface does not determine the complex: generic quadratic complexes form a four-dimensional moduli space, whereas their Kummer surfaces form a three-dimensional moduli space, and the generic cosingular family is one-dimensional. We give an explicit computational presentation of this classical missing modulus on the regular semisimple locus. If A is the Klein form, B is the second quadratic form, and T=A⁻¹B has six distinct eigenvalues, then the complex is determined by the unordered spectrum of T modulo affine transformations. Centering the spectrum gives a complete weighted-projective invariant [c₂:c₃:c₄:c₅:c₆]∈ℙ(2,3,4,5,6), where cⱼ are the elementary symmetric functions of the centered eigenvalues. Passing to the Kummer surface enlarges affine equivalence of the six spectral points to projective equivalence and forgets one marked-pencil parameter. We combine this quotient with exact focal computation. Restricting the second quadric to the plane of lines through a point produces a ternary conic; its determinant is the focal quartic, and its minors distinguish ordinary focal points from the sixteen nodes. Exact rank predicates, subresultants, and real-root isolation certify focal events along rational paths. The result provides a complete generic representation of quadratic line complexes and clarifies when such complexes can model structured ray families in visual computing.
MSC classes: Primary 14J28; Secondary 14M15, 14N05