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Isolated singularity and ruling comparison for quotients by non-central quadrics

Ascertain whether, for the algebra S and its three non-central linearly independent quadratic normal elements ω_i (1 ≤ i ≤ 3), the quotients A_i = S/ω_i are graded isolated singularities; and characterize the relationship between rulings defined via maximal Cohen–Macaulay modules and the geometric rulings (line-module families) described for S in Table 1.

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Background

Beyond the central quadratic case, S has several non-central quadratic normal elements ω_i. The paper shows A = S/z is a graded isolated singularity for central z, but leaves open whether similar properties hold for A_i = S/ω_i.

The question also seeks to compare homologically defined rulings via MCM modules against the geometric rulings furnishing line modules, probing the extent to which the two approaches coincide in the non-central setting.

References

Question The algebra S has three non-central linearly independent quadratic normal elements: \omega_i, 1 \leq i \leq 3. Let A_i = S/ w_i . Is A_i a graded isolated singularity? Referring to Table \ref{table:preimageLSofS}, the line modules corresponding to components, C_5, C_6 and C_{10} are annihilated by w_i, 1 \leq i \leq 3, respectively. What is the relationship between rulings defined via MCMs and the rulings given in Table \ref{table:preimageLSofS}?

— 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)