Completion of the proof for Family 2.14

Proof of the Family 2.14 assertion of Theorem 1.1. Suppose that a prime divisor $V$ over $X$ has $\beta_X(V)\le0$. Proposition 44.5 reduces its centre to a listed line, a point of such a line, or the node of a fibre of type $A_1$.

We first exclude every singular point of every fibre. Proposition 48.4, Corollary 49.3, and Theorem 51.1 give strict local threshold bounds at points of types $A_1,A_2,A_4$. At a point of type $A_3$, Lemmas 50.1 and 50.4 give $A_X(G)-S_X(G)=9/16$ and refined threshold at least one. Lemma 43.7 therefore gives $$\beta_X(V)\ge\frac9{16}\mathop{\mathrm{ord}}_V(G)>0$$ for every valuation centred at that point. The list in Lemma 43.2 contains no other singularity type.

It remains to exclude the listed lines and their smooth points on the fibre. Every such line meets the singular locus of its fibre, as follows from the same incidence configurations used in Proposition 44.5. A line on $V_5$ has normal bundle $\mathcal O\oplus\mathcal O$ or $\mathcal O(1)\oplus\mathcal O(-1)$; these are the ordinary lines and lines of the second kind, respectively, by Lemma 43.5. For an ordinary line the pencil differential consequently has rank one. Lemmas 45.2 and 45.3 exclude the line centre and every point on it where the fibre is smooth.

For a line of the second kind use its standard model $u=v=0$. A hyperplane containing it has equation $$\alpha x_1+\beta x_2+\gamma x_3+\delta x_4+\varepsilon x_6=0.$$ If $\alpha\ne0$, its differential in the positive normal direction never vanishes on the line, so the hyperplane section is smooth along it. This is impossible for a listed line. Hence $\alpha=0$. If $\delta=\beta=0$, normality of the fibre forces $\varepsilon\ne0$, and its closed-orbit point is of type $A_4$. Proposition 46.5 treats the rest of that line. If $\delta=0$ and $\beta\ne0$, the closed-orbit point is a node, and Proposition 46.6 treats the rest. Finally, if $\delta\ne0$, Proposition 47.3 treats every smooth point of the fibre on the line, including its intersection with $E$. These three cases exhaust the lines of the second kind. For a valuation whose centre is the whole line, choose a general smooth point of the fibre on that line and apply the corresponding local bound.

Every centre allowed by the reduction has now been excluded. Therefore $\beta_X(V)>0$ for every prime divisor over $X$, and the valuative criterion proves K-stability. ◻