Research Article · Preprint Mathematics
Sharp fourth-root concavity for lattice counts on smooth toric Fano threefolds
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- Mathematics
- Paper type
- Research Article
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- Preprint
- Manuscript
- Archived
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- 2 · Permanent link
Overview
Abstract
The fourth root of the ordinary Riemann–Roch count polynomial is concave on the entire closed nef cone of every smooth toric Fano threefold. The exponent 1/4 is uniformly optimal for these real polynomial extensions. Nine of the eighteen threefolds have concave cube root; the other nine have genuine cube-root failure. We prove a sufficient criterion for cubic full-count polynomials using classical Lorentzian volume theory and verify its hypotheses by exact fan and polynomial certificates. The result includes integral nonbig nef classes. For the split bundles P_{P²}(O ⊕ O(a)), the ordinary polynomial has optimal power 1/3 at twists a=0,1,2 and no positive concave power at any integer twist a≥3. For geometrically weighted lattice counts, biases r≥1 admit a nonpolynomial concave logarithmic extension for every twist, whereas every fixed 0<r<1 has integral midpoint counterexamples at large twists. The proof uses sharp coefficient conditions for general quadratic prefix weights and uniform rectangular-moment tail identities. It also gives exponential-quadratic extensions. No positive power works uniformly in the unweighted twist family, or at any fixed bias r>1 over prefixes. Ordinary-polynomial optimality and lattice-only sharpness are kept separate.
Keywords Algebra · Geometry
Scope of the work
Uniformly sharp fourth-root concavity of ordinary count polynomials on smooth toric Fano threefold nef cones, with exact eighteen-fan certificates and a separate weighted split-bundle logarithmic theory.
Reproduction notes
14-page manuscript and 8 package members. Compile main.tex with Tectonic or compatible LaTeX. With Python 3.9+ and its standard library, run verify_fano.py and verify_bundle.py in ordinary and optimized modes. The Fano checker reads the supplied 18-fan dataset and reconstructs the nef cones, count polynomials, pairwise inequivalence and 152 direct-count controls; the bundle checker verifies the exact weighted-prefix identities.
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Author disclosures
AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.
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Scott Schubr. (2026). Sharp fourth-root concavity for lattice counts on smooth toric Fano threefolds (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/concavity-of-lattice-counts-on-toric-threefolds/v/2
BibTeX
@techreport{luminary_concavity_of_lattice_counts_on_toric_threefolds_v2,
title = {Sharp fourth-root concavity for lattice counts on smooth toric Fano threefolds},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/concavity-of-lattice-counts-on-toric-threefolds/v/2}
}Loading discussion…