{
  "abstract": "Given a convex lattice polygon $Δ\\subset \\mathbb R^2$, let $N_Δ$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_Δ$ on the toric variety $\\mathbb P_Δ$ defined by $Δ$, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_Δ$. We perform calculations of $N_Δ$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-Göttsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06821",
  "author_total": 3,
  "authors": [
    {
      "name": "Tom Graber"
    },
    {
      "name": "Mingyuan Hu"
    },
    {
      "name": "Eric Zaslow"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0tkgs8vn8frdw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06821",
  "fulltext_url": null,
  "id": "0tkgs8vn8frdw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06821",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7211990496",
  "pdf_url": "https://arxiv.org/pdf/2609.06821",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0tkgs8vn8frdw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06821",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06821"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Mirror Counts of Spectral Curves",
  "title_text": "Mirror Counts of Spectral Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/0tkgs8vn8frdw",
  "venue": "arXiv",
  "versions": []
}