The statement for Family 3.2

Family 3.2: double conics in a projective bundle

We consider smooth Fano threefolds of Picard rank three and anticanonical degree fourteen. Belousov–Loginov [43] proved their K-stability under a condition on the cubic surface containing each double conic: its singularities along that conic must be ordinary double points. We use the corrected third version of their paper. In particular, its reduced-conic estimate is non-strict; the first adjunction step retains a positive discrepancy term, as recalled below.

The remaining double conics are treated by the normalization of the blowup of their fibre ideals. Its exceptional divisor $G$ has $A_X(G)=3$ and $S_X(G)=135/56$. We compute the actual refined series on $G$, including all cancellations. The different is one half of a smooth curve of bidegree $(2,1)$ and remains in the surface calculation. The refined threshold is at least one in the remaining cases. The positive difference $33/56$ then proves the theorem. No Zariski decomposition on the weighted threefold extraction is used. The volume decompositions on $X$ in the reduction below are retained.