{"paper":{"title":"Mirror Counts of Spectral Curves","arxiv_id":"2609.06821","primary_cat":"math.SG","submitted_at":"2026-09-06T20:29:18+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.99,"id":"s0","own":false,"role":"background","text":"Given a convex lattice polygon $\\Delta\\subset \\mathbb R^2$, let $N_\\Delta$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_\\Delta$ on the toric variety $\\mathbb P_\\Delta$ defined by $\\Delta$, having fixed intersection with the toric boundary.","hedge":{"p":0.54,"label":"asserted"},"own_p":0.13,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.99,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":1,"p":0.67,"id":"s1","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.31,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.67,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.32,"interpretation":0.0}},{"i":2,"p":0.53,"id":"s2","own":true,"role":"contribution","text":"In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\\Delta$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.53,"quantitative":false,"quantitative_p":0.12,"role_probabilities":{"gap":0.0,"method":0.04,"result":0.43,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.53,"interpretation":0.0}},{"i":3,"p":0.61,"id":"s3","own":true,"role":"method","text":"We perform calculations of $N_\\Delta$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.13,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.61,"result":0.08,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.31,"interpretation":0.0}},{"i":4,"p":0.9,"id":"s4","own":true,"role":"result","text":"Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.9,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.04}},{"i":5,"p":0.53,"id":"s5","own":false,"role":"interpretation","text":"We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-G\\\"ottsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau.","hedge":{"p":1.0,"label":"speculative"},"own_p":0.16,"main_claim_p":0.15,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.47,"contribution":0.0,"interpretation":0.53}}]},"compiled":{"main_claim":{"p":0.95,"text":"In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\\Delta$.","location":"abstract s2","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per 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