The line-closed matroid–quadratic Orlik–Solomon conjecture
Let be a matroid, and let be its Orlik–Solomon algebra. The matroid is line-closed if every subset closed under the closure of each pair of its elements is a flat, and is quadratic if it has a presentation whose relations all have degree two.
Line-closed matroid–quadratic Orlik–Solomon conjecture. is line-closed if and only if is quadratic.
The conjecture proposes a purely matroidal characterization of quadratic Orlik–Solomon algebras. The surrounding discussion identifies line-closure as a combinatorial analogue of formality, but the supplied text gives no resolution of the equivalence.
References
Primary source
Michael Falk, “Line-closed matroids, quadratic algebras, and formal arrangements”, arXiv:math/0010167 (2001).
Progress summary
A 2000 counterexample shows that the proposed equivalence is false, although one direction remains valid.
The conjecture was stated in a 1999 lecture. It asks whether line-closure exactly characterizes quadratic Orlik–Solomon algebras.
Known results
- Quadraticity implies line-closure over arbitrary coefficient fields.
2000 counterexample
Yuzvinsky found a rank-three matroid on with nontrivial lines , , , , and . It is line-closed but its Orlik–Solomon algebra is not quadratic: while . Thus line-closure is not sufficient, and the proposed equivalence is reported as disproved.
Current status (as of September 2026): Quadraticity implies line-closure, while Yuzvinsky's reported counterexample shows that line-closure does not imply quadraticity; the conjecture is therefore claimed disproved, pending independent verification.
Solutions 0
No solutions have been posted yet.