Good geometry for balanced quadratic tCY algebras

Establish that every quadratic d-dimensional twisted graded Calabi–Yau algebra with balanced underlying quiver has adequate geometry for d equal to 2 or 3, and prove that such an algebra has good geometry when its underlying quiver has at most three vertices.

Background

The conjecture is presented as a partial answer to Question 5.12. It concerns quadratic twisted graded Calabi–Yau algebras whose underlying quiver is balanced, meaning that each vertex has equal in-degree and out-degree.

The first assertion predicts that the strong point-stack projections are immersions. The second predicts equality of their images, and hence good geometry, when the quiver has at most three vertices. The authors note that their examples support the conjecture but that they have not analyzed all algebras on non-balanced quivers.

References

Conjecture 5.13. Let d ∈ {2, 3} and suppose A = kQ/I is a quadratic d-tCY algebra such that Q is balanced. Then A has adequate geometry. If moreover |Q0| ≤ 3, then A has good geometry.

— Point Modules of Non-Connected Graded Algebras  (2610.23171 - Lamkin, 16 Oct 2026) in Conjecture 5.13, Section 5.1, p. 54

In particular, we do not know how to determine whether Im πs1 = Im πs2 without doing example-by-example calculations.

— Point Modules of Non-Connected Graded Algebras  (2610.23171 - Lamkin, 16 Oct 2026) in Section 5.1, immediately before Example 5.11, p. 54