Completion of the proof for Family 3.2

Proof of Theorem 83.1. Suppose that $F$ is a prime divisor over $X$ with $\beta_X(F)\le0$. Proposition 85.4 places its centre in the reduced support $\ell$ of a double conic, disjoint from $D$. In the normal form of Lemma 84.2, suppose first that $g$ has distinct roots. Lemma 84.6 shows that the cubic fibre has exactly two singular points along $\ell$, both of type $A_1$. The local calculation [43, Lemma 7.1] applies to this double conic and gives $\delta_p(X)\ge56/45$ at every $p\in\ell\setminus D$, a contradiction. This invocation of the local lemma imposes no condition on the other double conics of $X$.

It remains to consider $g=r^2$ and $f=s(cr+ds)$, where $d\ne0$. Extract $G$ as in Lemma 84.4. Corollary 87.3 gives $\delta(G,\Gamma/2;W)\ge1$. The strict form of Abban–Zhuang adjunction yields $$\beta_X(F)\ge \bigl(A_X(G)-S_X(G)\bigr)\operatorname{ord}_F(G) =\frac{33}{56}\operatorname{ord}_F(G)>0.$$ Indeed, the centre on the extraction is contained in $G$, because $c_X(F)\subset\ell$. To see explicitly why a refined threshold equal to one suffices, take bases compatible with the $G$-filtration and $\operatorname{ord}_F$. Restrict the moving part of their pullbacks to $G$. Inversion of adjunction for the plt extraction and the refined threshold estimate give, upon taking limits, $$S_X(F)\le A_X(F)-A_X(G)\operatorname{ord}_F(G) +S_X(G)\operatorname{ord}_F(G).$$ This is precisely the displayed positive remainder, and contradicts the choice of $F$. ◻