On the gap between the integral and rational Gongyo indices of toric Fano varieties

Exact calculations for smooth toric Fano varieties in dimensions at most seven suggest a sharp dimension-dependent lower bound for the positive difference between the rational and integral Gongyo indices. We compute both indices for the smooth toric Fano models obtained from blow-ups of projective space at torus-invariant points by the anticanonical minimal model program. For pseudo-symmetric smooth toric Fano varieties, we prove the proposed lower bound and determine all equality cases. We also apply a dimension-raising construction to obtain an explicit infinite family of smooth toric Fano varieties with positive gap from odd-dimensional weak Fano varieties with zero gap. For the standard models considered, we determine the smallest dimension increase needed to obtain a Fano variety by iterating this construction and compute both indices of the minimal lifts. Finally, every positive rational number occurs as a Gongyo-index gap, even within the pseudo-symmetric class.