Dice Question Streamline Icon: https://streamlinehq.com

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).

Information Square Streamline Icon: https://streamlinehq.com

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)