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

@misc{sato2026between,
  title = {{On the gap between the integral and rational Gongyo indices of toric Fano varieties}},
  author = {Hiroshi Sato},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22997},
  eprint = {2609.22997},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22997},
}

TY  - UNPB
TI  - On the gap between the integral and rational Gongyo indices of toric Fano varieties
AU  - Hiroshi Sato
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22997
UR  - https://arxiv.org/abs/2609.22997
AB  - 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.
ER  -