Section 53

The conic bundle and the remaining fibres

Part IX · Family 2.16: conics on intersections of two quadricsPDF page 198

Family 2.16: conics on intersections of two quadrics

Let $V\subset\mathbb P^5$ be a smooth intersection of two quadrics, let $C\subset V$ be a smooth conic, and let $\pi\colon X=\mathop{\mathrm{Bl}}_CV\to V$ . These are the smooth members of Family 2.16. Put $H=\pi^*\mathcal O_V(1)$ , denote the exceptional divisor by $E$ , and set $B=H-E$ . Projection from the plane $\Pi=\langle C\rangle$ gives the conic bundle $g\colon X\to\mathbb P^2$ defined by $|B|$ . Thus $L=-K_X=H+B=2H-E$ and $L^3=22$ .

The estimates for reduced fibres are already available in [28, Propositions 5.2.3, 5.2.4 and 5.2.6]. The remaining fibres are double lines. We blow up their scheme-theoretic fibre ideals, obtaining an exceptional surface $F\cong\mathbb P^1\times\mathbb P^1$ with different $D/2$ , where $D$ is a smooth curve of bidegree $(2,1)$ . The new calculation identifies its refined sections with powers of the ideal of one point on $D$ . This is simpler than the length-two condition in Family 2.13, but the different must still be retained.

Theorem 53.1. Every smooth member $X$ of Family 2.16 satisfies $\delta(X)\ge176/171$ . More precisely, if $p$ lies on a nonreduced conic fibre, then $\delta_p(X)\ge88/85$ .

Lemma 53.2 (Intersection numbers and three expected orders). The intersection numbers are $H^3=4$ , $H^2E=0$ , $HE^2=-2$ , and $E^3=-2$ . Moreover, $S_X(H)=\frac4{11},\qquad S_X(E)=\frac{23}{44},\qquad S_X(B)=\frac{13}{22}.$ Every prime divisor on $X$ has positive beta invariant.

Proof. The blow-up formulas give the intersection numbers: in particular, $\deg N_{C/V}=(-K_V)C+2g(C)-2=2$ . Both $H$ and $B$ are nef. If $T\ne E$ is a prime divisor and $T\sim aH+bB$ , intersection with a general conic fibre and a general fibre of $E\to C$ gives $a,b\ge0$ . Consequently, the effective cone is generated by $E$ and $B$ , and $(aH-bE)^3=4a^3-6ab^2+2b^3.$ The divisors $L-uH$ and $L-uE$ are nef for $0\le u\le1$ and are not pseudoeffective for $u>1$ . Their volume integrals are $\begin{aligned} S_X(H)&=\frac1{22}\int_0^1 \bigl(4(2-u)^3-6(2-u)+2\bigr)\,du=\frac4{11},\\ S_X(E)&=\frac1{22}\int_0^1 \bigl(32-12(1+u)^2+2(1+u)^3\bigr)\,du=\frac{23}{44}. \end{aligned}$ For $L-uB$ , the pseudoeffective threshold is two. The divisor is nef on $[0,1]$ . On $[1,2]$ its fixed part is $(u-1)E$ , and the moving part is $(2-u)H$ . Indeed, intersection with the moving fibres of $E\to C$ forces that multiplicity, and the remaining divisor is semiample. Thus $S_X(B)=\frac1{22}\left( \int_0^1\bigl(4(2-u)^3-6(2-u)(1-u)^2+2(1-u)^3\bigr)\,du +\int_1^2 4(2-u)^3\,du\right)=\frac{13}{22}.$ For $T\ne E$ , the integers $a,b$ above are not both zero, so $T-H$ or $T-B$ is effective. Monotonicity of volumes gives $S_X(T)\le13/22<1=A_X(T)$ . The calculation for $E$ treats the other prime divisor. ◻

Proposition 53.3 (The known reduced-fibre estimates). If the conic fibre through $p$ is reduced, then $\delta_p(X)\ge176/171$ .

Proof. We use the three cited propositions of [28]. For a smooth fibre, unless it is the negative section of $E\cong\mathbb F_2$ , they give $176/161$ off $E$ and $176/169$ on $E$ . For a reduced singular fibre, Proposition 5.2.4 gives $176/171$ , including points on $E$ and the intersection of its two components. In the exceptional smooth-fibre case, Proposition 5.2.6 gives $22/19$ . Their minimum is $176/171$ . These are local estimates for all divisors whose centres contain $p$ ; no automorphism-group assumption occurs in these propositions. ◻

53.1 A section ring adapted to the projection

Choose equations $Q_1,Q_2$ of $V$ so that $Q_1$ contains $\Pi$ and $Q_2|_\Pi$ is the equation of $C$ . On $\Pi$ use three coordinates, and use $t_0,t_1,t_2$ transverse to $\Pi$ . In the blow-up of $\Pi$ in $\mathbb P^5$ , write $t_i=e y_i$ , with $[y_0:y_1:y_2]\in\mathbb P^2$ . The variable $e$ has divisor class $E=H-B$ .

Lemma 53.4 (The projective-bundle presentation). Let $T=\mathbb P_{\mathbb P^2}(\mathcal O^{\oplus3}\oplus\mathcal O(1))$ , in the quotient convention, with tautological class $H$ and base class $B$ . Then $T=\mathop{\mathrm{Bl}}_\Pi\mathbb P^5$ , and $X\subset T$ is the complete intersection $F_1=Q_1/e=0,\qquad F_2=Q_2=0, \qquad \deg F_1=H+B,\quad\deg F_2=2H.$ For $a,b\ge0$ , its spaces $H^0(X,aH+bB)$ are the $(a,b)$ pieces of $R=\frac{\mathbb C[s,x,z,e,y_0,y_1,y_2]}{(F_1,F_2)},\qquad \begin{array}{c|ccc} \text{variable}&s,x,z&e&y_0,y_1,y_2\\\hline (H,B)\text{-degree}&(1,0)&(1,-1)&(0,1). \end{array}$ In particular, $N_m=h^0(X,mL)=\frac{(2m+1)(11m^2+11m+6)}6.$

Proof. The three coordinates on $\Pi$ and the products $e y_i$ give the six sections of $H$ , identifying $T$ with the graph of the projection. The restrictions of $Q_1,Q_2$ give the displayed equations. Near each point of $C$ , one coefficient of a transverse variable in $Q_1$ is a unit: otherwise that point would be singular on $Q_1$ and hence on $V$ . Locally eliminate that variable. The incidence equations then become the usual graph of the blow-up of the remaining regular codimension-two ideal. Away from $C$ the graph is already $V$ . This proves the complete-intersection assertion, including along the exceptional divisor.

For $a\ge0$ , the push-forward of $\mathcal O_T(aH+bB)$ is $\bigoplus_{d=0}^a \operatorname{Sym}^{a-d}(\mathbb C^3)\otimes \mathcal O_{\mathbb P^2}(b+d),$ and the higher direct images vanish. Thus its sections are exactly the polynomial piece with the stated degrees. Twists with $H$ -coefficient $-1,-2,-3$ are acyclic on the projective-space fibres. In the Koszul resolution of $X$ , the additional twists of $aH+bB$ have degrees $(a-1,b-1)$ , $(a-2,b)$ , and $(a-3,b-1)$ . Whenever their $H$ -coefficient is nonnegative, their push-forwards have base twists at least $-1$ , hence no higher cohomology. Otherwise they are among the three acyclic relative twists just listed. Taking sections therefore gives the asserted quotient. Its Hilbert function on the diagonal, or Riemann–Roch with $L^3=22$ , gives $N_m$ . ◻