AI 中文总结
研究实超平面排列中过滤的Varchenko - Gelfand代数的克雷莫纳不变性，通过特定运算证明其在交换系数环上的同构，还展示了反例反驳从该代数重建顶图的猜想。
AI 中文摘要
我们证明了过滤的Varchenko - Gelfand代数在一类实超平面排列的自然克雷莫纳运算下是不变的。假设一个排列包含所有坐标超平面，且其余每个定义形式都支撑在两个坐标上。在每个这样的形式中交换两个系数可得到其克雷莫纳变换。坐标反转给出腔室双射以及在每个交换系数环上相应过滤的Varchenko - Gelfand代数的同构。作为应用，我们展示了\(\mathbb{R}^3\)中两个具有同构过滤的Varchenko - Gelfand代数但非同构顶图的八个中心平面的排列。这反驳了Yagi - Yoshinaga关于从过滤的Varchenko - Gelfand代数重建顶图的猜想。
英文摘要
We prove that the filtered Varchenko--Gelfand algebra is invariant under a natural Cremona operation on a class of real hyperplane arrangements. Namely, suppose that an arrangement contains all coordinate hyperplanes and that every remaining defining form is supported on two coordinates. Swapping the two coefficients in each such form produces its Cremona transform. Coordinatewise inversion gives a chamber bijection and an isomorphism of the corresponding filtered Varchenko--Gelfand algebras over every commutative coefficient ring. As an application, we exhibit two arrangements of eight central planes in $\mathbb{R}^3$ with isomorphic filtered Varchenko--Gelfand algebras but non-isomorphic tope graphs. This disproves a conjecture of Yagi--Yoshinaga on reconstructing tope graphs from filtered Varchenko--Gelfand algebras.
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