Section 68

The local threshold and the proof for Family 2.17

Part X · Family 2.17: line extractions and weighted pointsPDF page 244

Lemma 68.1 (The local threshold at a triple-contact point). For every $p\in\Sigma_X$ , one has $\delta_p(X)\ge\frac{64}{59}>1.$

Proof. Let $\sigma:\widetilde X\to X$ be the weight- $(1,1,2)$ blow-up in Lemma 66.2, with exceptional divisor $G$ . This is a plt extraction. Indeed, its two weight-one charts are smooth, and its weight-two chart is the quotient of affine three-space by $\boldsymbol\mu_2$ acting with weights $(1,1,1)$ . The divisor $G$ is $\mathbb P(1,1,2)$ . Restricting the index-one cover of the weight-two threefold chart gives $\mathbb A^2\to\mathbb A^2/\boldsymbol\mu_2$ . The action on this surface cover is free away from the origin, so the different on $G$ is zero. The spaces in Lemma 66.5 are the actual refinement of the anticanonical series along $G$ . Lemma 66.6 verifies its bounded support and ample-series hypotheses. Abban–Zhuang adjunction (Proposition 2.7, or [4, Theorem 3.3]), Lemma 66.4, and Lemma 67.8 give $\delta_p(X)\ge \min\left\{\frac{A_X(G)}{S_X(G)},\delta(G;W)\right\} \ge\min\left\{\frac{32}{29},\frac{64}{59}\right\} =\frac{64}{59}.$ This is a lower bound for $\delta_p(X)$ ; no equality is asserted. ◻

Proof of Theorem 61.1. Suppose that a prime divisor $V$ over $X$ has $\beta_X(V)\le0$ . Proposition 61.2 puts its centre at a point $p\in\Sigma_X$ . By Lemma 68.1, $A_X(V)\ge\frac{64}{59}S_X(V)>S_X(V),$ a contradiction. Hence every prime divisor over $X$ has positive beta invariant. The valuative criterion proves K-stability of every smooth member of Family 2.17. ◻

Remark 68.2 (No generality assumption). The coefficient $\nu$ was only required to be nonzero; its nonvanishing follows from the exact double contact along the first contracted line. The second initial spaces are independent of its nonzero value. The prime initial divisor and the exact Hilbert count also account for arbitrary higher-weight terms of the original equation. Thus the estimate applies to every point of $\Sigma_X$ on every smooth member, without an assumption on the remaining coefficients.

Remark 68.3 (Why no threefold Zariski decomposition is needed here). We have not computed a Zariski decomposition on the weighted point blow-up. The replacement is an exact calculation of its section filtration. The integral initial divisor in the symplectic flag variety identifies the associated graded ring with the actual weighted point refinement. The second degeneration is an automorphism degeneration on $\mathbb P(1,1,2)$ . Its complete binary spaces account for every section, as the exact Hilbert count shows. Their first moments give both the expected order and the refined threshold.

The threefold volume decompositions in the earlier line extractions remain part of the reduction to $\Sigma_X$ . At the final weighted extraction, however, neither an assumed nef chamber nor a discarded higher-order term enters the argument. In particular the failing line surface estimate $49/24>2$ is retained: the weighted point refinement is a different filtration, and its identification is proved before Abban–Zhuang theory is applied.