Section 42

Completion of the proof for Family 2.13

Part VII · Family 2.13: genus-two sextics on quadricsPDF page 163

Proof of Theorem 38.1 and the Family 2.13 assertion of Theorem 1.1. Every fibre of $\psi$ is a nonzero plane conic. It is consequently smooth, a reduced union of two distinct lines, or a double line. The reduced-fibre estimates established above give $80/77$ on a smooth fibre, $64/61$ away from the node of a reducible fibre, and $160/159$ at its node. Theorem 41.10 gives $320/319$ at every point of a double fibre. These four cases exhaust $X$ .

All these estimates concern the local threshold defined in the common preliminaries. In particular they control every prime divisor over $X$ whose centre contains the chosen point. For any prime divisor $V$ over $X$ , choose $p\in c_X(V)$ . Then $\frac{A_X(V)}{S_X(V)}\ge\delta_p(X)\ge\frac{320}{319}>1.$ Taking the infimum over $V$ gives $\delta(X)\ge320/319$ , and the valuative criterion proves K-stability. ◻

The numerical inputs are kept separate from the completeness assertions. The original threefold contributes the elementary fixed-part formula (39.1); the reduced fibres use explicit Zariski decompositions on surfaces. For double fibres, Lemma 41.3 identifies the actual associated ring, and Lemma 41.4 identifies its restricted spaces. The subsequent enlargement is controlled by both their orders and their dimensions. No Zariski decomposition on the weighted threefold extraction is used. The missing dimension in the limiting spaces is retained precisely because discarding it would lose the strict inequality needed in the proof.