Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement
Abstract
We disprove Sato's weak F-equivalence conjecture for nonsingular projective toric weak Fano varieties in every dimension d ≥ 3. Our counterexamples are smooth projective crepant models of centered reflexive simplices. The key input is a rigidity property of ray polytopes: if XΣ is nonsingular and complete and −KXΣ is nef, then every nonzero lattice point of PΣ=Conv(G(Σ)) is a primitive ray generator. For our models, this rules out every weak-Fano-preserving equivariant blow-up and blow-down throughout the flop class. We then introduce Gorenstein weak F-equivalence, generated by projective toric birational zigzags through normal projective Gorenstein toric weak Fano varieties, and formulate a corresponding refinement of Sato's conjecture. We prove this refined conjecture in dimensions d ≤ 3, as well as for the family of counterexamples constructed above in every dimension. Finally, we show that the refined conjecture implies the inclusion-connectivity of reflexive d-polytopes modulo unimodular equivalence, which is known for d ≤ 4 and remains open for d ≥ 5. The results were developed with the assistance of GPT-5.6 Sol.
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BibTeX
@misc{chakravarty2026counterexamples,
title = {{Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement}},
author = {Avik Chakravarty and Daebeom Choi and Shengjing Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18054},
eprint = {2608.18054},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18054},
}
RIS
TY - UNPB
TI - Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement
AU - Avik Chakravarty
AU - Daebeom Choi
AU - Shengjing Xu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2608.18054
UR - https://arxiv.org/abs/2608.18054
AB - We disprove Sato's weak F-equivalence conjecture for nonsingular projective toric weak Fano varieties in every dimension d ≥ 3. Our counterexamples are smooth projective crepant models of centered reflexive simplices. The key input is a rigidity property of ray polytopes: if X_Σ is nonsingular and complete and −K_X_Σ is nef, then every nonzero lattice point of P_Σ=Conv(G(Σ)) is a primitive ray generator. For our models, this rules out every weak-Fano-preserving equivariant blow-up and blow-down throughout the flop class. We then introduce Gorenstein weak F-equivalence, generated by projective toric birational zigzags through normal projective Gorenstein toric weak Fano varieties, and formulate a corresponding refinement of Sato's conjecture. We prove this refined conjecture in dimensions d ≤ 3, as well as for the family of counterexamples constructed above in every dimension. Finally, we show that the refined conjecture implies the inclusion-connectivity of reflexive d-polytopes modulo unimodular equivalence, which is known for d ≤ 4 and remains open for d ≥ 5. The results were developed with the assistance of GPT-5.6 Sol.
ER -