Optimal concavity powers for lattice counts on smooth toric Fano threefolds

Abstract

We determine the optimal concavity power of the ordinary Riemann–Roch count polynomial on the entire closed nef cone of every smooth toric Fano threefold. Nine of the eighteen types have optimal power 1/3, two have optimal power 1/4, and the remaining seven have explicit rational optimal powers strictly between these values. In particular, 1/4 is the sharp uniform exponent. A Lorentzian coefficient criterion proves the uniform theorem; exact coefficientwise determinant certificates and origin or nef-ray curvature determine the individual optima. Integral nonbig nef classes are included. For the split projective bundles ℙ(𝒪 ⊕ 𝒪(a)) over ℙ², 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. Sharp coefficient conditions for general quadratic prefix weights, uniform rectangular-moment tail identities and exponential-quadratic extensions supply this second proof route. No positive power works uniformly in the unweighted twist family, or at any fixed bias r>1 over prefixes. The ordinary-polynomial classification and lattice-only Jensen assertions concern distinct extensions.