Exclusion of all centres and completion of the proof

Proof. Let $C$ be its centre. A ruling of $E\to\ell$ meets the negative section $s$, which lies in $f^{-1}(y_0)$. If this fibre is reduced, Lemma 25.1 excludes such a centre; if it is nonreduced, Corollary 28.6 does so.

Otherwise $C\sim as+bg$ with integers $a\ge1$ and $b\ge0$, so $C-s$ is linearly equivalent to an effective divisor. Monotonicity of the surface volume and Lemma 26.2 give $$S(W^E;C)\le S(W^E;s)=\frac{71}{72}<1.$$ At the generic point of $C$, the only divisorial valuation of the surface $E$ is a positive multiple of $\operatorname{ord}_C$. Thus its refined threshold there is at least $72/71$. Abban–Zhuang adjunction along the smooth divisor $E$, with $S_X(E)=5/9$, gives $$\frac{A_X(Q)}{S_X(Q)}\ge \min\left\{\frac95,\frac{72}{71}\right\}>1$$ for every prime divisor $Q$ with centre $C$. This proves the assertion. ◻

Proof. For the refinement of the surface series by $F$, Lemma 24.2 gives $\mathcal S_F=29/48$ and $\mathcal T_p=1$. Applying surface adjunction first shows that the threshold of $W^S$ at $p$ is at least one. Now $S_X(S)=41/72<1$, so the strict adjunction inequality of Lemma 21.2 gives the displayed lower bound. Since the centre is $p\in S$, the order of $S$ is positive. ◻

Proof of Theorem 23.1. Suppose first that $\ell$ is of the first kind, and let $Q$ be a prime divisor over $X$. If its centre is contained in $E$, Lemma 26.1 gives $\beta_X(Q)>0$. Otherwise choose a point of its centre outside $E$. According as its conic fibre is smooth, reduced and singular, or nonreduced, Lemma 24.2, Lemma 24.3, or Corollary 28.6 gives a local threshold strictly greater than one. Thus $\beta_X(Q)>0$ in every case, proving K-stability for first-kind lines.

Now suppose that $\ell$ is of the second kind. Let $Q$ be any prime divisor over $X$. If its centre is a divisor on $X$, Lemma 24.1 gives $\beta_X(Q)>0$.

Suppose next that its centre is a curve. If the curve is contained in $E$, apply Lemma 30.1. Otherwise take a point of the curve outside $E$. For a smooth conic fibre, Lemmas 23.4 and 24.2 give local threshold at least $9/8$. For a reduced singular fibre away from $y_0$, Lemma 24.3 gives at least $36/35$; for the reduced fibre over $y_0$, use Lemma 25.1. For a nonreduced fibre, Corollary 28.6 gives at least $144/127$. In every case this threshold is strictly greater than one and therefore excludes $\beta_X(Q)\le0$.

It remains to consider a point centre $p$. If $p\notin E$, the same four estimates apply. If $p\in E$ lies on a singular conic fibre, the reduced-fibre lemmas and the double-fibre corollary again apply. The only case left is a point of $E$ on a smooth conic. Such a point is outside $s$, because the fibre containing $s$ is singular. Choose $S$ as in Lemma 29.1. If $(cF)_p=1$, Lemma 30.2 gives strict positivity. If $(cF)_p=2$, the coordinate normalization of Lemma 29.1 permits application of Lemma 29.7, which gives a local threshold strictly greater than one. Thus $\beta_X(Q)>0$ for every prime divisor $Q$ over $X$. The valuative criterion proves K-stability. ◻