Section 11

Expected vanishing orders on the threefold

Part III · Family 2.9: conic bundlesPDF page 51

We use the invariants defined in Section 2, with $L^3=16$ . The distinction between divisors on $X$ and divisors over $X$ is retained throughout.

Lemma 11.1. One has $H^3=1,\quad H^2E=0,\quad HE^2=-7,\quad E^3=-36, \qquad H^2B=3,\quad HB^2=2,\quad B^3=0.$ The nef cone is generated by $H,B$ , and the effective cone by $E,B$ .

Proof. The blow-up formulas give the first three intersections. The degree of $N_{C/\mathbb P^3}$ is $4\deg C+2g(C)-2=36$ , giving $E^3=-36$ . Substituting $B=3H-E$ gives the remaining intersections and $(H+B)^3=16$ .

Both $H$ and $B$ are nef. If $f$ is a fibre of $E\to C$ , then $Hf=0$ and $Bf=1$ . If $z$ is a general conic fibre, then $Hz=2$ and $Bz=0$ . As the Picard rank is two, these tests give the asserted nef cone.

Let $D\ne E$ be a prime divisor. It is the strict transform of a surface of degree $a\ge1$ with multiplicity $b\ge0$ along $C$ , so $D\sim aH-bE$ . Restrict to a general smooth cubic $S$ not contained in $D$ . The restriction is effective, and its intersection with the nef conic class $F$ is $2a-6b$ . Thus $a\ge3b$ . It follows that $D=(a-3b)H+bB$ , with nonnegative coefficients. Since $H=(E+B)/3$ , all effective classes lie in the cone generated by $E,B$ , and both generators are effective. ◻

Lemma 11.2 (The three volume integrals). The volume functions needed in Theorem 10.1 are $\begin{aligned} \mathop{\mathrm{vol}}(L-uH)&=(1-u)^3+9(1-u)^2+6(1-u),&&0\le u\le1,\\ \mathop{\mathrm{vol}}(L-uE)&=16-60u+24u^2+36u^3,&&0\le u\le\tfrac13,\\ \mathop{\mathrm{vol}}(L-uB)&= \begin{cases}1+9(1-u)+6(1-u)^2,&0\le u\le1,\\ (4-3u)^3,&1\le u\le\tfrac43. \end{cases} \end{aligned}$ They vanish beyond the displayed intervals.

Proof. The effective cone gives the thresholds $1,1/3,4/3$ , respectively. Up to the first two thresholds the divisors are nef: $L-uH=(1-u)H+B,\qquad L-uE=(1-3u)H+(1+u)B.$ Likewise $L-uB=H+(1-u)B$ is nef for $0\le u\le1$ . The cube of $aH+bB$ is $a^3+9a^2b+6ab^2$ .

For $1\le u\le4/3$ , write $L-uB=(4-3u)H+(u-1)E.$ The $E$ term is fixed: restriction to a general fibre of $E\to C$ forces its coefficient, and the remaining complete series is pulled back from $\mathbb P^3$ . Equivalently, $\pi_*\mathcal O_X(kE)=\mathcal O_{\mathbb P^3}$ for $k\ge0$ . The volume is therefore $(4-3u)^3$ . ◻

Proof of Theorem 10.1. Integration of Lemma 11.2, with denominator $16$ , gives $S_X(H)=\frac1{16}\left(\frac14+3+3\right)=\frac{25}{64},\quad S_X(B)=\frac1{16}\left(\frac{15}{2}+\frac1{12}\right)=\frac{91}{192},$ $S_X(E)=\frac1{16}\left(\frac{16}{3}-\frac{10}{3}+\frac8{27}+\frac19\right) =\frac{65}{432}.$ If $D\ne E$ is prime, put $k=a-3b$ as in Lemma 11.1. The integers $k,b$ are nonnegative and not both zero. If $k\ge1$ , then $D-H$ is effective and $S_X(D)\le S_X(H)$ . If $k=0$ , then $b\ge1$ , $D-B$ is effective and $S_X(D)\le S_X(B)$ . These inequalities follow from monotonicity of volume. Since every prime divisor on the smooth threefold has discrepancy one, all these beta invariants are positive, including that of $E$ . ◻