Proof for every smooth member

Proof of Theorem 108.1. Let $Q$ be a prime divisor over $X$. Suppose that $\beta_X(Q)\le0$. Proposition 110.6 puts its centre on one of the two lines through the singular point of an $A_1$ or $A_2$ fibre. Denote this line by $\Gamma$ and choose a closed point $p$ of the centre.

If $p$ is the singular point of the fibre, Proposition 112.1 and Corollary 113.6 give $\delta_p(X)>1$, a contradiction. If $p\in E$, use Lemma 114.3 or Proposition 114.6. Their bounds are again strictly greater than one.

Assume that $p$ is outside these two points. Interchange the plane projections so that the first contracts $\Gamma$. If $p\notin D_2$, Lemma 111.10 supplies a smooth quintic del Pezzo section through $p$ on which $p$ lies on none of the ten lines. Lemmas 111.5, 111.7 and 111.8, with the two projections interchanged, give $\delta_p(X)\ge1152/1127>1$. If $p\in D_2$, Lemma 111.10 says that the fibre has type $A_1$ and $p$ is its remaining smooth corner. Proposition 115.1 gives $\delta_p(X)>1$ there. This is the last possibility.

In every case $A_X(Q)>S_X(Q)$, contrary to our assumption. Thus every prime divisor over $X$ has positive beta invariant. The valuative criterion proves K-stability. ◻