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  "title": "A sharp smoothness criterion for bi-Lagrangian Grassmannians",
  "shortTitle": "When two forms give a smooth space",
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  "oneLine": "The reduced parameter space is smooth exactly when every Jordan pencil block has dimension two.",
  "abstract": "A written proof candidate settles the sharp smoothness part of AMR-020-0502. Kronecker blocks impose no restriction; longer Jordan blocks create a singular fixed-locus obstruction. The single-block singularity and proposed criterion are prior work. Orbit classification, incidence-scheme reducedness and nonlinear integrability remain outside the result.",
  "datePublished": "2026-09-12",
  "dateModified": "2026-09-12",
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  "keywords": [
    "bi-Lagrangian Grassmannian",
    "alternating pencil",
    "Jordan-Kronecker decomposition",
    "smoothness",
    "integrable systems"
  ],
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    "Smoothness of the reduced complex variety is equivalent to dimension-two Jordan pencil blocks; Kronecker blocks are unrestricted.",
    "The smooth variety is a product of ordinary Lagrangian Grassmannians.",
    "A length-n block has a distinguished chart A^(n-2) times {a^n+cz=0}.",
    "The same semisimplicity criterion governs real algebraic smoothness and the embedded real smooth locus being a manifold."
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    "Classify the full singular locus for mixed Jordan blocks.",
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  "relatedWorks": [
    {
      "citation": "Bolsinov, Matveev, Miranda and Tabachnikov (2018), Problem 5.3: the motivating smoothness and orbit questions.",
      "url": "https://doi.org/10.1098/rsta.2017.0430",
      "doi": "10.1098/rsta.2017.0430"
    },
    {
      "citation": "Kozlov (2024), Geometry of bi-Lagrangian Grassmannian, Sections 3 and 13, Problem 14.8 and Remark 14.4.",
      "url": "https://arxiv.org/abs/2409.09855v1",
      "doi": "10.48550/arXiv.2409.09855"
    },
    {
      "citation": "Malkin, Ostrik and Vybornov (2005), Section 5.1.1: classical single-block affine-Schubert singularities.",
      "url": "https://arxiv.org/abs/math/0305095",
      "doi": null
    },
    {
      "citation": "Pappas and Zhou (2025), Theorem 1.1: affine-Schubert smooth loci at absolutely special vertices.",
      "url": "https://doi.org/10.1007/s00208-025-03123-8",
      "doi": "10.1007/s00208-025-03123-8"
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