Section 101

Completion of the smooth classification

Part XIII · Family 3.5: K-semistability and the complete smooth classificationPDF page 342

Theorem 101.1. If $\eta$ has no totally ramified point, then $\delta(X)\ge\frac{100}{99}>1.$ Every smooth member of Family 3.5 is K-semistable. It is K-stable if and only if $\eta$ has no totally ramified point. It is K-polystable if and only if $\eta$ has either no totally ramified point or two such points. In the latter case $\eta$ is equivalent to $[x:y]\mapsto[x^5:y^5]$ .

Proof. Assume first that $\eta$ has no totally ramified point. The proof of Theorem 96.4 gives the bound $100/99$ at every point except, possibly, points on the joining lines of mixed fibres. Their $A_2$ endpoints are covered by Theorem 94.1. Their $A_1$ endpoints and smooth interior points are covered by Theorems 98.1 and 100.1. All three bounds $160/151$ , $240/221$ and $32/29$ exceed $100/99$ . Thus the same lower bound holds at every closed point of $X$ . Every divisorial valuation has a centre containing a closed point, so the valuative criterion gives $\delta(X)\ge100/99$ and K-stability.

The totally ramified cases were proved in Section 91: one such point gives strict K-semistability, whereas two give the monomial K-polystable member. There cannot be more than two, by Riemann–Hurwitz. The monomial member has a nontrivial one-parameter automorphism group, so is not K-stable. This proves every assertion. ◻

The weighted curve extraction gives another proof of semistability at the interior points. Its last point moment is exactly one; the ordinary point extraction is needed here for the stronger K-stability statement. In both arguments, no Zariski decomposition on the extracted threefold is assumed or computed. The complete associated series are identified from actual sections by exact Hilbert counts. The subsequent estimates concern complete binary spaces and explicit surface resolutions.