Relate Smith–Van den Bergh rulings to geometric rulings for S

Characterize the relationship between the Smith–Van den Bergh ruling(s) on Proj A, with A = S/(x_1^2 + x_2^2 + x_3 x_4 + x_4 x_3), and the geometric rulings producing line modules described for S via the components C_i of its line scheme and their associated quadrics (as summarized in Table 1).

Background

For S, the paper identifies ten conic components of the line scheme and associates to each a smooth quadric in P3, with a distinguished ruling that yields line modules. Independently, Smith–Van den Bergh define rulings using indecomposable MCM modules.

The question asks for a precise correspondence or comparison between these two notions of ruling, potentially aligning module-theoretic and geometric viewpoints in the setting of the quotient by the central quadratic element.

References

Question Referring to Table \ref{table:preimageLSofS}, the line modules corresponding to component C_9 are all annihilated by the central element z = x_12+x_22+x_3x_4+x_4x_3 \in S. Hence they are line modules for the quotient A = S/ z . What is the relationship between the notion of ruling defined by and the ruling given in Table 1?

— Some Artin-Schelter Regular Algebras From Dual Reflection Groups and their Geometry (2410.08959 - Goetz et al., 11 Oct 2024) in Section 6.3 (Graded isolated singularities and rulings)