Determine the generic isolated singularities of P4L

Determine the ideal and corresponding isolated singularities arising from the intersection of the varieties \(\mathbf{V}(\mathcal{H})\) and \(\mathbf{V}(\mathcal{K})\) for four generic observed lines, after removing the singularity loci consisting of the observed lines and their transversal lines.

Background

For P4L, the rank-deficiency conditions of the interaction matrix are represented by polynomial ideals. The component $\mathbf{V}(\mathcal{G})$ is fully characterized geometrically as the locus of camera centers on transversal lines intersecting the four observed lines. The remaining component, $$\mathbf{V}(\mathcal{H}) \cap \mathbf{V}(\mathcal{K})$$, is intended to describe the isolated singularities after the observed lines and their transversals are removed through saturation.

The paper derives the saturation formulation for this remaining variety but does not compute it symbolically for generic line parameters. Numerical specializations suggest that, generically, there are at most ten isolated complex singularities, with some potentially real; a complete generic derivation and geometric interpretation remain unresolved.