Section 61

K-stability of Family 2.17

Part X · Family 2.17: line extractions and weighted pointsPDF page 214

Family 2.17: line extractions and weighted points

Let $\pi:X=\mathop{\mathrm{Bl}}_C(\mathbb P^3)\longrightarrow\mathbb P^3, \qquad q:X=\mathop{\mathrm{Bl}}_{C'}(Q)\longrightarrow Q$ be the two contractions of a smooth member of Family 2.17. Here $C$ is a smooth elliptic quintic, $Q\subset\mathbb P^4$ is a smooth quadric, and $C'$ is a smooth elliptic normal quintic. Write $H=\pi^*\mathcal O_{\mathbb P^3}(1),\quad B=q^*\mathcal O_Q(1),\quad E=\operatorname{Exc}(\pi),\quad E'=\operatorname{Exc}(q),\quad D=E\cap E',\quad L=-K_X.$ The intersection defining $D$ is scheme-theoretic. The standard relations are $\label{f217:eq:classes} B=3H-E,\quad H=2B-E',\quad E'=5H-2E,\quad E=5B-3E', \qquad L=H+B,\qquad L^3=24.$ Cheltsov–Pokora [40, Main Theorem] proved that a prime divisor over $X$ with nonpositive beta invariant has centre a closed point of $D$ . We complete the point analysis and prove the following theorem.

Theorem 61.1. Every smooth Fano threefold in Family 2.17 is K-stable. At each point of the triple-contact set $\Sigma_X$ defined in (65.1), one has $\delta_p(X)\ge64/59$ .

We first prove the reduction supplied by the two line extractions. The final weighted point calculation then treats its remaining centres.

Proposition 61.2. If $V$ is a prime divisor over $X$ with $\beta_X(V)\le0$ , its centre belongs to $\Sigma_X$ . In particular, $X$ is K-stable if $\Sigma_X=\varnothing$ .

Remark 61.3 (The two refinements). Smoothness of $X$ does not imply smoothness of $D$ . At the remaining triple-contact points, the first line refinement has expected order $49/24$ at a surface point blow-up of discrepancy two. We retain this calculation and use a different extraction. Its actual weighted section ring gives $S_X(G)=29/8$ and refined threshold at least $64/59$ . The proof of Theorem 61.1 is completed in Section 68.

Lemma 61.4. The divisor $D$ is a reduced curve. Its maps to $C$ and $C'$ are finite flat of degrees two and three. Through a point of $D$ the two contracted ruling lines have distinct tangent directions. The map $(\pi,q):X\to\mathbb P^3\times Q$ is a closed immersion, so $L$ is very ample.

Proof. For a fibre $\ell$ of $E\to C$ , one has $H\ell=0$ , $E\ell=-1$ , and $B\ell=1$ . Thus $q|_\ell$ is an isomorphism onto a line. This line cannot be contained in the elliptic curve $C'$ , so $\ell$ is not contained in $E'$ . Similarly a fibre $\ell'$ of $E'\to C'$ has $H\ell'=1$ and cannot be contained in $E$ . Since $E'\ell=2$ and $E\ell'=3$ , the stated maps are finite of degrees two and three.

Write $D=\sum n_iD_i$ as a divisor on $E$ . Every component dominates both elliptic curves. If its two degrees are $a_i,b_i$ , then $\sum n_i a_i=2$ and $\sum n_i b_i=3$ . A multiplicity $n_i\geq2$ would force $D=2D_i$ , contradicting the second equality. Thus $D$ is generically reduced. It is a Cartier divisor on the smooth surface $E$ , so has no embedded points and is reduced. Its structure sheaf is torsion free over each smooth base curve and hence flat.

The tangent direction of $\ell'$ is killed by $dq$ , whereas $dq$ is nonzero on the tangent direction of $\ell$ . They are therefore distinct. The same argument shows that $(\pi,q)$ has injective differential everywhere: off $E$ the first map is an isomorphism, and on $E$ its kernel is the ruling direction, on which $dq$ is injective. It is injective on geometric points because $q$ embeds every $\pi$ -fibre. This proper, universally injective, unramified map is a closed immersion. The last assertion follows by restricting $\mathcal O(1,1)$ . ◻