A Universal Obstruction to the Samuelson Condition for Tangent Lagrangian 2-Webs
Abstract
We study the Samuelson area condition for planar Lagrangian 2-webs formed by tangent lines. For the tangent line family Lt:y=tx+h(t), where h is smooth and h''≠0, let Ψ(u,v)=Lu∩ Lv for u≠ v, and let JΨ be its Jacobian. Domitrz and Zubilewicz (2024) proved that these webs do not satisfy the Samuelson condition. We study the Jacobian as the two tangents coalesce. In web coordinates, the Samuelson condition is equivalent to ∂u∂vlog|JΨ|=0. We show that JΨ has a simple zero along u=v and that ∂u∂vlog|JΨ(u,v)| =1/((v−u)2)+Rh(u,v), where Rh extends smoothly across the diagonal. The double pole therefore prevents the Samuelson condition from holding near the envelope. We also give a direct local proof of this failure for webs formed by two separated smooth line families with varying slopes.
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BibTeX
@misc{kurokawa2026universal,
title = {{A Universal Obstruction to the Samuelson Condition for Tangent Lagrangian 2-Webs}},
author = {Yasuhiro Kurokawa},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.22260},
eprint = {2606.22260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22260},
}
RIS
TY - UNPB
TI - A Universal Obstruction to the Samuelson Condition for Tangent Lagrangian 2-Webs
AU - Yasuhiro Kurokawa
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2606.22260
UR - https://arxiv.org/abs/2606.22260
AB - We study the Samuelson area condition for planar Lagrangian 2-webs formed by tangent lines. For the tangent line family L_t:y=tx+h(t), where h is smooth and h''≠0, let Ψ(u,v)=L_u∩ L_v for u≠ v, and let J_Ψ be its Jacobian. Domitrz and Zubilewicz (2024) proved that these webs do not satisfy the Samuelson condition. We study the Jacobian as the two tangents coalesce. In web coordinates, the Samuelson condition is equivalent to ∂_u∂_vlog|J_Ψ|=0. We show that J_Ψ has a simple zero along u=v and that ∂_u∂_vlog|J_Ψ(u,v)| =1/((v−u)^2)+R_h(u,v), where R_h extends smoothly across the diagonal. The double pole therefore prevents the Samuelson condition from holding near the envelope. We also give a direct local proof of this failure for webs formed by two separated smooth line families with varying slopes.
ER -