Cremona invariance of filtered Varchenko--Gelfand algebras

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.