Weak Lefschetz Property for codimension-three Gorenstein algebras

Determine whether every Artinian Gorenstein algebra A = k[x_1, x_2, x_3]/I in codimension three has the Weak Lefschetz Property, i.e., whether there exists a linear form ℓ such that ×ℓ: [A]_i → [A]_{i+1} has maximum rank in all degrees.

Background

The Weak Lefschetz Property is a central topic in the study of graded algebras. While WLP is known in several settings (e.g., for certain complete intersections and monomial cases), its validity for all codimension-three Artinian Gorenstein algebras remains unsettled.

The authors emphasize that this question is still open, highlighting the broader landscape of unresolved Lefschetz property problems beyond the specific conic-focused results developed in the paper.

References

In fact, for Gorenstein algebras k[x_1,x_2,x_3]/I, even the WLP is an open question.

— The non-Lefschetz locus of conics  (2404.16238 - Marangone, 2024) in Section 8 (General Gorenstein Algebras), concluding paragraph

So, codimension three is exactly the case where the answer is not known, and it has been conjectured that in characteristic zero every Artinian Gorenstein algebra of codimension three has the WLP (see, e.g.,). In characteristic zero, the question has been settled in several special cases, but it remains open in general.

— On the weak Lefschetz property of Artinian Gorenstein algebras of codimension three in arbitrary characteristic  (2608.27232 - Javadekar, 27 Aug 2026) in Section 1, Introduction