Concavity of Lattice Counts on Toric Threefolds
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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.
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Research details
- Subjects
- Mathematics · Algebra · Geometry
- Keywords
- Algebra, Geometry
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
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 concerns the ordinary real polynomial.
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