Research Article · Preprint Mathematics
Concavity of Lattice Counts on Toric Threefolds
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- Mathematics
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- Research Article
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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. This exponent is uniformly optimal for the ordinary real polynomial extensions. Precisely nine of the eighteen threefolds have concave cube root; the other nine have cube-root failure, certified by exact fan and polynomial computations. Integral nonbig nef classes are included. For split projective-line bundles over the projective plane, the ordinary polynomial has optimal power one third at twists zero, one and two, while no positive concave power exists at any integer twist at least three. Geometrically weighted lattice counts with bias at least one admit nonpolynomial concave logarithmic extensions at every twist, whereas every fixed bias strictly between zero and one has integral midpoint counterexamples at large twists. Ordinary-polynomial optimality and lattice-only sharpness are distinguished.
Keywords Algebra · Geometry
Scope of the work
The fourth root of the ordinary Riemann-Roch count polynomial is concave on the entire closed nef cone of every smooth toric Fano threefold. This exponent is uniformly optimal for the ordinary real polynomial extensions. Precisely nine of the eighteen threefolds have concave cube root; the other nine have cube-root failure, certified by exact fan and polynomial computations. Integral nonbig nef classes are included. For split projective-line bundles over the projective plane, the ordinary polynomial has optimal power one third at twists zero, one and two, while no positive concave power exists at any integer twist at least three. Geometrically weighted lattice counts with bias at least one admit nonpolynomial concave logarithmic extensions at every twist, whereas every fixed bias strictly between zero and one has integral midpoint counterexamples at large twists. Ordinary-polynomial optimality and lattice-only sharpness are distinguished. Quarter optimality is ordinary-real-polynomial, not lattice-only; no classification of fixed-twist lattice powers, arbitrary weakFano/smooth threefolds, higher-dimensional TU or universal interpolation.
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The approved 8-member package contains the complete 14-page paper, editable LaTeX sources and dependencies, reader/build instructions, scholarly references and a public checksum manifest, with necessary exact verification programs and certificates. Follow the included README and retain the package together. Exact finite certificates support the stated analytic argument; they do not certify external human peer review.
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Scott Schubr. (2026). Concavity of Lattice Counts on Toric Threefolds (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/concavity-of-lattice-counts-on-toric-threefolds/v/1
BibTeX
@techreport{luminary_concavity_of_lattice_counts_on_toric_threefolds_v1,
title = {Concavity of Lattice Counts on Toric Threefolds},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/concavity-of-lattice-counts-on-toric-threefolds/v/1}
}Loading discussion…