Sylvester characterization of linearly presented graded Möbius algebras

Determine whether, for every simple matroid with at least three elements whose graded Möbius algebra is quadratic, the graded Möbius algebra is linearly presented if and only if the matroid is Sylvester.

Background

The paper studies when the defining ideal of the graded Möbius algebra of a simple matroid is quadratic and linearly presented. A matroid is Sylvester when every pair of elements belongs to a 3-element circuit. The authors verify that the conjecture holds for uniform matroids of rank two and for several small affine and projective geometries.

The authors prove the implication from quadratic and linearly presented graded Möbius algebras to Sylvester matroids in Proposition 4.2. The converse implication—that every Sylvester matroid whose graded Möbius algebra is quadratic has a linearly presented graded Möbius algebra—remains unresolved.

References

We conjecture (see \Cref{conj:Sylv_iff_lin_pres}) that for matroids with at least 3 elements for which $GMA{M}$ is quadratic, $I_M$ has only linear first syzygies, i.e. $GMA{M}$ is linearly presented, if and only if $M$ is Sylvester, meaning every two elements of $M$ belong to a unique circuit of size $3$.

— Graded Betti numbers of graded Möbius algebras of uniform matroids  (2609.03827 - Fouli et al., 3 Sep 2026) in Conjecture 1.1, Section 4 (Linearly presented graded Möbius algebras)