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Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement · arXivDesk

2608.18054Aug 18, 202618 pages. Comments are welcome

Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement

Avik Chakravarty, Daeboem Choi, Shengjing Xu

Abstract

We disprove Sato's weak $F$ -equivalence conjecture for nonsingular projective toric weak Fano varieties in every dimension $d \geq 3$ . 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 $-K_{X_Σ}$

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is nef, then every nonzero lattice point of

P_Σ=\operatorname{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 \leq 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 \leq 4

and remains open for

d \geq 5

. The results were developed with the assistance of GPT-5.6 Sol.

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