佐藤弱F等价猜想的反例与一个戈伦斯坦改进
Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement
AI总结：
该研究构造了所有维度d≥3的佐藤弱F等价猜想反例，提出戈伦斯坦弱F等价及对应改进猜想，证明其在d≤3及反例维度成立，还揭示其与自反单形包含连通性的关联。
AI中文摘要：
我们对所有维度d≥3的非奇异射影环面弱法诺（weak Fano）簇，否定了佐藤（Sato）弱F等价猜想。我们的反例是中心自反单形的光滑射影亏格收缩（crepant）模型。关键输入是射线多面体的刚性性质：若X_Σ非奇异且完备，且-K_{X_Σ}是数值有效的（nef），则P_Σ=Conv(G(Σ))的每个非零格点都是本原射线生成元。对于我们的模型，这排除了整个翻转（flop）类中所有保持弱法诺的等变胀开（blow-up）与胀缩（blow-down）。随后，我们引入戈伦斯坦弱F等价，其由通过正规射影戈伦斯坦环面弱法诺簇的射影环面双有理之字形（birational zigzag）生成，并提出佐藤猜想的对应改进版本。我们证明该改进猜想在维度d≤3时成立，且对上述构造的所有维度反例均成立。最后，我们表明该改进猜想蕴含自反d-单形在幺模等价下的包含连通性，该性质在d≤4时已知成立，而d≥5时仍为开放问题。本研究成果借助GPT-5.6 Sol协助完成。
英文摘要：
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 \(-K_{X_Σ}\) 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.
