Section 89

Geometry and the scope of the result

Part XIII · Family 3.5: K-semistability and the complete smooth classificationPDF page 306

Family 3.5: K-semistability and the complete smooth classification

We prove that every smooth member of Family 3.5 is K-semistable and classify the K-stable and K-polystable members. The mixed fibre requires two additional point extractions. Their actual refined series are identified from sections and exact Hilbert counts; explicit resolutions of conic and cubic pencils then control every divisor over the exceptional surfaces. A weighted extraction of the joining line gives an independent semistability estimate.

Put $M=\mathbb P^1\times\mathbb P^2$ . Let $Q\subset\mathbb P^2$ be a smooth conic, and identify $S=\mathbb P^1\times Q$ with $\mathbb P^1\times\mathbb P^1$ by $([u:v],[x:y])\longmapsto([u:v],[y^2:xy:x^2]).$ Choose relatively prime binary quintics $f,g$ . The curve $C=\{u f(x,y)+v g(x,y)=0\}\subset S$ is the graph of a degree-five morphism $\eta:C\cong\mathbb P^1\longrightarrow\mathbb P^1$ . Thus $C$ is smooth even when $\eta$ has high ramification. Conversely, every smooth member of Family 3.5 is obtained as $\pi:X=\operatorname{Bl}_C(M)\longrightarrow M$ in this way; see [3, Sections 5.14 and 7.5]. The bidegree $(5,2)$ of $C$ in $M$ means that the degrees of the two pulled-back hyperplane bundles on $C$ are five and two.

Write $T=\pi^*\mathcal O_M(1,0)$ , $H=\pi^*\mathcal O_M(0,1)$ , let $E$ be the exceptional divisor, and let $D$ be the strict transform of $S$ . We use $T$ also for a fibre of $\phi:X\to\mathbb P^1$ . Then $D=2H-E,\qquad L=-K_X=2T+3H-E,\qquad L^3=20.$ On $D\cong\mathbb P^1\times\mathbb P^1$ , let $\ell_1,\ell_2$ denote the pullbacks of points of its first and second factors. Notice the factor two in the restriction of $H$ : $T|_D=\ell_1,\quad H|_D=2\ell_2,\quad E|_D=\ell_1+5\ell_2,\quad D|_D=-\ell_1-\ell_2,\quad L|_D=\ell_1+\ell_2.$ The intersection identities on $X$ are $TH^2=1,\quad TE^2=-5,\quad HE^2=-2,\quad E^3=-14;$ products with one $E$ and two pullbacks vanish, as do $T^2$ and $H^3$ . Indeed, $\deg N_{C/M}=(-K_M)\cdot C-2=14$ . These identities give $L^3=54-48+14=20$ .

Definition 89.1. The ramification index of $\eta$ at $q\in C$ is the integer $r$ for which, in local coordinates, $\eta$ has the form $t=c x^r+O(x^{r+1})$ , with $c\ne0$ . A ramification point is simple if $r=2$ , and is totally ramified if $r=5$ . The ramification divisor is $R=\sum_q(r_q-1)q$ ; the Riemann–Hurwitz formula gives $\deg R=8$ .

Theorem 89.2 (The smooth Family 3.5 classification). Let $X$ be a smooth member of Family 3.5. Then $X$ is K-semistable. More precisely,

If $\eta$ has no totally ramified point, then $\delta(X)\ge100/99$ and $X$ is K-stable.
If $\eta$ has a totally ramified point, then $X$ is K-semistable. If it has two such points, then $X$ is the K-polystable member corresponding to $u x^5+v y^5=0$ . If it has exactly one, then $X$ is strictly K-semistable.

The first assertion extends Denisova’s simple-ramification theorem [14]; its proof is completed in Theorem 101.1. The second is [3, Corollary 5.71 and Lemma 7.6]; we give the degeneration and the requisite Abban–Zhuang argument below. The local estimates at index-three and index-four points are respectively $\delta_p(X)\ge160/151$ and $\delta_p(X)=400/389$ . For the mixed fibre, the index-two endpoint and the interior of its joining line satisfy $\delta_p(X)\ge240/221$ and $\delta_p(X)\ge32/29$ , respectively. The announced work [15] is not used as an unpublished theorem.