The Nash manifold of four-point configurations modulo similarity subgroups

Abstract

We construct and study the Nash manifold of four-point configurations in the real plane modulo the action of a Nash subgroup of the group of similarity transformations. We do so by using the finer notion of a quadrangle in place of that of a four-point configuration, since this retains limiting line data in degenerations. We prove that the space 𝒬 of quadrangles is an 8-dimensional Nash manifold and that, for every Nash subgroup G of the similarity group, the orbit space 𝒬/G is a Nash manifold. We define a geometric invariant, called aspect, with values in [−1,1], and prove that, over each of the intervals (−1,0) and (0,1), the corresponding part of 𝒬/G is Nash diffeomorphic to the product of the interval with a fixed fiber. Thus the nonexceptional part of the moduli problem reduces to the analysis of two model fibers, each having a natural geometric interpretation in terms of the quadrangles themselves.

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BibTeX RIS

BibTeX

@misc{olberding2026nash,
  title = {{The Nash manifold of four-point configurations modulo similarity subgroups}},
  author = {Bruce Olberding and Elaine A. Walker},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02087},
  eprint = {2610.02087},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02087},
}

RIS

TY  - UNPB
TI  - The Nash manifold of four-point configurations modulo similarity subgroups
AU  - Bruce Olberding
AU  - Elaine A. Walker
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02087
UR  - https://arxiv.org/abs/2610.02087
AB  - We construct and study the Nash manifold of four-point configurations in the real plane modulo the action of a Nash subgroup of the group of similarity transformations. We do so by using the finer notion of a quadrangle in place of that of a four-point configuration, since this retains limiting line data in degenerations. We prove that the space 𝒬 of quadrangles is an 8-dimensional Nash manifold and that, for every Nash subgroup G of the similarity group, the orbit space 𝒬/G is a Nash manifold. We define a geometric invariant, called aspect, with values in [−1,1], and prove that, over each of the intervals (−1,0) and (0,1), the corresponding part of 𝒬/G is Nash diffeomorphic to the product of the interval with a fixed fiber. Thus the nonexceptional part of the moduli problem reduces to the analysis of two model fibers, each having a natural geometric interpretation in terms of the quadrangles themselves.
ER  -

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