Fat point modules from ruling-two lines for S and T

Ascertain whether, for each component C_i (1 ≤ i ≤ 10) of the line scheme of the algebras S or T, every line ℓ lying in the second ruling ℛ_{i,2} of the associated smooth quadric Q_i in P^3 and not contained in any ruling-one family ℛ_{j,1}, yields a truncated module F(ℓ) = [M(ℓ)(1)]_{≥1} that is always a fat point module of multiplicity two over the corresponding algebra (S or T).

Background

For the AS regular algebras S and T constructed in the paper, each line-scheme component C_i corresponds to a smooth quadric Q_i in P^3, with two rulings; only one ruling produces line modules. The other ruling empirically appears to produce fat point modules of multiplicity two.

The question asks to verify this phenomenon uniformly for all ruling-two lines not in any ruling-one family, establishing a systematic connection between these geometric lines and fat point modules.