Section 39

Expected vanishing and quartic sections

Part VII · Family 2.13: genus-two sextics on quadricsPDF page 145

Lemma 39.1. The expected vanishing orders are $S_X(H)=\frac38,\qquad S_X(E)=\frac{73}{320}, \qquad S_X(B)=\frac{41}{80}.$ Moreover, every prime divisor $D$ on $X$ has $\beta_X(D)>0$ .

Proof. For $a,b\ge0$ set $V(a,b)=(aH+bB)^3=2a^3+12a^2b+6ab^2.$ The effective cone gives $\tau(H)=1$ and $\tau(E)=1/2$ . The corresponding divisors remain nef up to their thresholds. Therefore $\begin{aligned} S_X(H)&=\frac1{20}\int_0^1 (20-36u+18u^2-2u^3)\,du=\frac38,\\ S_X(E)&=\frac1{20}\int_0^{1/2} (20-48u+6u^2+20u^3)\,du=\frac{73}{320}. \end{aligned}$ For $B$ the threshold is $3/2$ . Up to $u=1$ , the divisor $L-uB=H+(1-u)B$ is nef. For $1\le u\le3/2$ its moving and fixed parts are $\label{f213:eq:Bparts} P(u)=(3-2u)H,\qquad N(u)=(u-1)E.$ Indeed, intersection with the moving exceptional fibres forces the stated multiplicity of $E$ , and the residual divisor is a basepoint-free pullback. Consequently $S_X(B)=\frac1{20}\left( \int_0^1(20-24u+6u^2)\,du+ \int_1^{3/2}2(3-2u)^3\,du\right)=\frac{41}{80}.$ For a prime $D\ne E$ , Lemma 38.5 gives $D\sim aH+bB$ with nonnegative integral coefficients, not both zero. Thus $D-H$ or $D-B$ is effective. Monotonicity of volume gives $S_X(D)\le41/80<1=A_X(D)$ . The displayed value for $E$ gives the remaining assertion. ◻

Lemma 39.2. Let $y\in\mathbb P^2$ have reduced conic fibre $F_y$ . A general line $\ell$ through $y$ has smooth inverse image $S=\psi^{-1}(\ell)$ . The restriction $\pi|_S$ is an isomorphism onto a smooth intersection of two quadrics in $\mathbb P^4$ , and $C_S=E|_S$ is a smooth curve isomorphic to $C$ . Writing $A=H|_S=-K_S,\qquad F=B|_S,$ we have $A^2=4,\quad A\cdot F=2,\quad F^2=0, \qquad C_S=2A-F,\quad C_S^2=8,\quad A\cdot C_S=6.$

Proof. Bertini gives smoothness away from $F_y$ . At every smooth point of this fibre, $d\psi$ has rank two. If the fibre is reducible, it has one node, where $d\psi$ has rank one because $X$ is smooth. Only one tangent direction of $\ell$ is forbidden there. Thus a general choice is smooth everywhere.

The map $\psi$ sends every exceptional fibre $e_c\subset E$ isomorphically to a line, since $B\cdot e_c=1$ . These lines form a curve $\Gamma$ in the dual plane, and $\Gamma$ is contained in no line. Otherwise all $\psi(e_c)$ would pass through a fixed point $z$ , so both linear forms $M(x)z$ would vanish on the nondegenerate curve $C$ , and hence identically. This contradicts their independence. The pencil of lines through $y$ is therefore not contained in $\Gamma$ , and a general $S$ contains no $e_c$ .

Eliminating $y\in\ell$ in the two incidence equations shows that $\pi(S)$ is the intersection of $Q$ with a quadric containing $C$ . Locally write $C=(a,b)$ and the latter equation as $\alpha a+\beta b+O((a,b)^2)=0.$ Its strict transform contains the whole fibre $e_c$ precisely when $\alpha(c)=\beta(c)=0$ . This was excluded; consequently $\pi(S)$ is smooth along $C$ , and $C$ is Cartier on it. Its blow-up along $C$ is trivial, so $\pi|_S$ is an isomorphism. Away from $C$ the map is already an isomorphism. The curve $C_S$ is therefore isomorphic to $C$ . Adjunction and Lemma 38.5 give the final formulas. ◻

Example 39.3. A reduced conic node can lie on $E$ . For the first matrix in (38.1), take $Q=\{x_0x_3+x_1^2+x_2^2+x_4^2=0\}.$ The fibre over $[0:0:1]$ has equation $x_1^2+x_2^2=0$ in the plane $x_3=x_4=0$ . Its node $c=[1:0:0:0:0]$ belongs to $C$ . The quadric is smooth. In scroll coordinates its intersection with the scroll is $s^3ab+t^2(s^2+t^2)a^2+t^2b^2=0.$ The coefficients never vanish simultaneously on $\mathbb P^1$ , and its discriminant $s^6-4s^2t^4-4t^6$ has six distinct roots. Hence $C$ is smooth. Lemma 39.2 includes this configuration; no assumption that the node avoids $E$ is needed.

Lemma 39.4. Let $S$ be as in Lemma 39.2, and refine the anticanonical series along $S$ . If $s_C$ is a defining section of $C_S$ , then its actual restricted spaces are $\label{f213:eq:actualW} W^S_{m,j}=\begin{cases} H^0(S,mA+(m-j)F),&0\le j\le m,\\ s_C^{\,j-m}H^0(S,(3m-2j)A),&m\le j\le\lfloor3m/2\rfloor,\\ 0,&j>3m/2. \end{cases}$ For every divisorial valuation $v$ over $S$ , $\label{f213:eq:refinedS} S(W^S;v)=\frac3{20}\int_0^1(4+4t)S_{A+tF}(v)\,dt +\frac3{40}S_A(v)+\frac1{80}v(C_S).$ Here $S_D(v)$ denotes expected vanishing for the complete series of the ample $\mathbb Q$ -divisor $D$ on $S$ , extended continuously to real coefficients.

Proof. For $j\le m$ , the divisor $mL-jB$ is $mH+(m-j)B$ . Its restriction map to $S$ is surjective: by Kawamata–Viehweg vanishing, $H^1(X,mH+(m-j-1)B)=0,$ since adding $-K_X$ gives the nef and big divisor $(m+1)H+(m-j)B$ . For $j\ge m$ , equation (39.1) gives the exact fixed part $(j-m)E$ , with residual series $|(3m-2j)H|$ . This residual series also restricts surjectively: for $d\ge0$ , $H^1(X,dH-B)=0, \qquad dH-B-K_X=(d+1)H$ is nef and big. The effective threshold excludes larger $j$ . This proves (39.2); in particular these are equalities of spaces, not merely containments.

For $u=j/m$ , the normalized moving and fixed divisors on $S$ are $\begin{array}{c|c|c|c} u& M(u)&N(u)|_S&M(u)^2\\ \hline {[0,1]}&A+(1-u)F&0&8-4u\\ {[1,3/2]}&(3-2u)A&(u-1)C_S&4(3-2u)^2. \end{array}$ The denominator $N_m\sim(20/6)m^3$ and surface Riemann–Roch give density $(3/20)M(u)^2\,du$ . The fixed contribution is exactly $\frac3{20}\int_1^{3/2}4(3-2u)^2(u-1)\,du\;v(C_S) =\frac1{80}v(C_S).$ The moving contribution on the second interval is $\frac3{20}\int_1^{3/2}4(3-2u)^3\,du\;S_A(v) =\frac3{40}S_A(v).$ Changing variables to $t=1-u$ in the first interval proves (39.3). These limits also follow directly from the finite adapted bases in (39.2). The support is bounded, and the spaces contain an ample series, so the basis limits used in the common preliminaries apply. ◻

Remark 39.5. The coefficient $1/80$ in (39.3) is the contribution of the fixed components. Vanishing along $C_S$ arising from the moving sections is already included in the two $S$ -terms. Since $C_S$ is smooth, $v(C_S)\le A_S(v)$ for every divisorial valuation $v$ . This remains true at a conic node lying on $C_S$ .