Schwarzian Residue of the Samuelson Obstruction in Tangent Lagrangian 2-Webs

Abstract

The Samuelson condition, a classical area-ratio condition for planar Lagrangian $2$-webs, is equivalent in local web coordinates $(u,v)$ to $\partial_u\partial_v\log|J_F(u,v)|=0$, where $F$ is the inverse web-coordinate map and $J_F$ is its Jacobian. For the tangent-line family $L_t:y=tx+h(t)$, let $\Psi(u,v)$, for $u\neq v$, be the intersection map of $L_u$ and $L_v$, write $J_\Psi$ for its Jacobian, and set $S_h(u,v):=\partial_u\partial_v\log|J_\Psi(u,v)|$. Near each diagonal point $(t_0,t_0)$ with $h''(t_0)\neq0$, the obstruction admits the decomposition $S_h(u,v)=1/(v-u)^2+R_h(u,v)$, where $R_h$ extends smoothly across the diagonal near $(t_0,t_0)$. We call $R_h(t,t)$ the Schwarzian residue and compute $R_h(t,t)=h^{(4)}(t)/(3h''(t))-(4/9)(h'''(t)/h''(t))^2=(1/2)\{\sigma,t\}=-\kappa_{\mathrm{aff}}(\sigma)(d\sigma/dt)^2$. Here $\{\sigma,t\}$ denotes the Schwarzian derivative of $\sigma$ with respect to $t$, where $\sigma$ is the equi-affine arclength parameter of the envelope $\gamma(t)=(-h'(t),h(t)-th'(t))$, and $\kappa_{\mathrm{aff}}$ denotes its equi-affine curvature, with the convention $\gamma_{\sigma\sigma\sigma}+\kappa_{\mathrm{aff}}\gamma_\sigma=0$. Thus the universal pole accounts for the local failure of the Samuelson condition, while the diagonal finite part carries affine-projective information about the envelope. We also derive the transformation law of the Schwarzian residue under reparametrization of the tangent-line parameter.