A Universal Obstruction to the Samuelson Condition for Tangent Lagrangian 2-Webs

Abstract

We study the Samuelson area condition for Lagrangian \(2\)-webs in the symplectic plane generated by tangent lines to plane curves. We prove that, near every point of nonzero curvature of a \(C^\infty\) regular plane curve, the local tangent \(2\)-web formed by nearby distinct tangent lines is not a Samuelson web. The obstruction comes from a universal local phenomenon: the Jacobian of the intersection map of two tangent lines has a simple zero along the diagonal where the two lines coalesce, producing a logarithmic singularity and hence a nonzero mixed derivative. This gives a direct local obstruction to the Jacobian characterization of the Samuelson condition. We also prove an analogous non-Samuelson result for separated real analytic tangent families whenever the associated intersection map defines a local \(2\)-web. These results show that the obstruction previously found by explicit computations for non-degenerate real conics is a manifestation of a general local mechanism for tangent families.