Section 103

Geometry and the statement for Family 3.6

Part XIV · Family 3.6: double conics and weighted curve extractionsPDF page 351

Family 3.6: double conics and weighted curve extractions

Let $C\subset\mathbb P^3$ be a smooth elliptic quartic, let $\Lambda\subset\mathbb P^3$ be a line disjoint from $C$ , and let $\pi:X\to\mathbb P^3$ blow up $C\sqcup\Lambda$ . These are exactly the smooth members of Family 3.6. Write $E,R$ for the exceptional divisors over $C,\Lambda$ , respectively, and put $H=\pi^*\mathcal O_{\mathbb P^3}(1),\qquad A=H-R,\qquad B=2H-E.$ The two pencils define morphisms $\sigma:X\to\mathbb P^1$ and $\phi:X\to\mathbb P^1$ . Their general fibres are del Pezzo surfaces of degrees five and six, respectively. The product $\eta=(\sigma,\phi)$ is a conic bundle over $\mathbb P^1\times\mathbb P^1$ . We prove the following assertion.

Theorem 103.1. Every smooth Fano threefold in Family 3.6 is K-stable.

Cheltsov proved this when every conic fibre is reduced [42, Theorem 1]. The omitted configurations are quintic fibres with two $A_1$ points or one $A_3$ point. We first reduce possible nonpositive-beta centres to their double conics. We then blow up the scheme-theoretic conic fibre. The resulting exceptional surface has a nonzero different, which is retained in every subsequent estimate.

Lemma 103.2. Write $C=(q_0=q_1=0)$ and $\Lambda=(l_0=l_1=0)$ . The variety $X$ is the complete intersection $u_0q_1-u_1q_0=0,\qquad v_0l_1-v_1l_0=0$ in $\mathbb P^3\times\mathbb P^1\times\mathbb P^1$ , of classes $2H+B$ and $H+A$ . In particular, $-K_X=H+A+B=4H-E-R,\qquad (-K_X)^3=22.$ Every quintic fibre is normal and has only Du Val singularities. Its possible singularity types are $A_1,A_2,2A_1,A_3$ .

Proof. Each equation is the graph of a pencil whose base ideal is a regular sequence. Since the two centres are disjoint, the product of the two graphs is their simultaneous blow-up. Adjunction gives the asserted anticanonical class. The nonzero intersection numbers needed below are $H^3=1,\quad HE^2=-4,\quad E^3=-16,\quad HR^2=-1,\quad R^3=-2;$ also $H^2E=H^2R=ER=0$ . They give the degree $22$ .

A plane through $\Lambda$ cuts out on the smooth curve $C$ a curvilinear scheme of length four. On that plane the fibre is the blow-up of the scheme. A length- $k$ part has local ideal $(y,x^k)$ ; one chart of its blow-up has equation $y t=x^k$ and gives an $A_{k-1}$ singularity. The partitions of four give the stated list. The pencil of restricted quadrics supplies a conic bundle on each such fibre. The partitions $(2,2)$ and $(4)$ give a double conic, as described explicitly below. ◻

Lemma 103.3 (The two fibre moments). For a fibre $S\in|A|$ and a fibre $T\in|B|$ , $S_X(S)=\frac{67}{88},\qquad S_X(T)=\frac{109}{176}.$ More precisely, the positive and negative parts of $-K_X-uS$ are $\begin{array}{c|c|c|c} u&P(u)&N(u)&P(u)^3\\ \hline {[0,1]}&(4-u)H-E+(u-1)R&0&22-15u\\ {[1,2]}&(4-u)H-E&(u-1)R&(4-u)^3-12(4-u)+16, \end{array}$ and those of $-K_X-uT$ are $\begin{array}{c|c|c|c} u&P(u)&N(u)&P(u)^3\\ \hline {[0,1]}&(4-2u)H+(u-1)E-R&0&22-18u\\ {[1,3/2]}&(4-2u)H-R&(u-1)E&(4-2u)^3-3(4-2u)+2. \end{array}$ The pseudoeffective thresholds are $2$ and $3/2$ , respectively.

Proof. The nef classes $H,A,B$ express the positive parts in the displayed intervals. For example, in the second interval of the first table, $P(u)=(2-u)H+B$ ; in that of the second table, $P(u)=(3-2u)H+A$ . Intersection with the fibres of $R\to\Lambda$ or $E\to C$ forces the indicated fixed part when $u>1$ . The effective cone is generated by $E,R,A,B$ [42, Section 2]; it gives the two thresholds. The intersection numbers in Lemma 103.2 give the cubics in the tables. Thus, explicitly, $\begin{aligned} 22S_X(S)&=\int_0^1(22-15u)\,du+ \int_1^2\big((4-u)^3-12(4-u)+16\big)\,du=\frac{67}{4},\\ 22S_X(T)&=\int_0^1(22-18u)\,du+ \int_1^{3/2}\big((4-2u)^3-3(4-2u)+2\big)\,du=\frac{109}{8}. \end{aligned}$ ◻

Lemma 103.4 (Previously established local estimates). The following estimates hold for the actual fibre refinements.

If $S\in|A|$ is smooth, then $\delta_p(S;W^S)>1$ for every $p\in S$ .
If $S\in|A|$ has just one singularity, of type $A_1$ or $A_2$ , then $\delta_p(S;W^S)>1$ away from the conic fibre through that singularity.
If $T\in|B|$ is smooth, then $\delta_p(T;W^T)\ge1$ . The inequality is strict unless $p\in E$ and $p$ lies on a $(-1)$ -curve of $T$ .

Every prime divisor whose centre on $X$ is a surface has positive beta.

Proof. The three estimates are [42, Corollaries 2–4]; they are local statements about the indicated fibre and do not assume that other fibres of $X$ are reduced. The divisorial assertion is the divisorial-stability calculation recalled in [42, Section 2]. ◻