Section 17

A uniform bound on the exceptional plane

Part III · Family 2.9: conic bundlesPDF page 67

We now complete the alternative exceptional-plane proof at the smooth-fibre points of $f_0$ , including arbitrary contact between $C$ and the conic. Throughout this section, $p\in f_0$ and $Z=\varphi^{-1}(\varphi(p))$ is smooth. We retain the two blow-ups and the divisors $R,T$ from Section 15. We do not assume (16.1).

Lemma 17.1 (A forced plane component without transverse-position assumptions). Blow up the strict transform of $Z$ on $Y_2$ , and write $U$ for the exceptional divisor and $Q$ for the strict transform of the plane surface $\pi^{-1}(\Pi)$ . On this smooth threefold, $U\cong\mathbb P^1\times\mathbb P^1,\qquad Q=H-2G-R-2T-U.$ For $t\ge5/2$ , every effective representative of the pullback of $D_t$ contains $\label{f29:eq:uniform-forced} \frac{t-1}{2}R+(t-1)T+(t-2)U+(2t-5)(Q+U).$ Moreover, $H^0(X,mL\otimes\mathfrak m_p^j)=0$ for $j>3m$ , and the pseudo-effective threshold of $G$ is exactly three.

Proof. The curve $Z$ is a smooth fibre, so $N_{Z/X}\cong\mathcal O^{\oplus2}$ . After blowing up $p$ , its normal bundle is $\mathcal O(-1)^{\oplus2}$ . It is disjoint from the centres producing $R,T$ , by Lemma 15.2. Thus its blow-up has exceptional quadric $U$ , with $U|_U=\mathcal O(-1,-1)$ .

In the local coordinates of that lemma, the plane surface has equation $\eta z+a(t)=0,\qquad a(t)\in(t^2).$ It has multiplicity two at $p$ . Its multiplicities along the subsequent centres give one copy of $R$ and two of $T$ . For example, after blowing up $p$ , its equation in the $z$ -chart is $\eta_1+z^{-2}a(zt_1)=0$ ; it contains $f$ , and its transform after blowing up $f$ contains the negative section. Along $Z$ its generic multiplicity is one. These facts prove the class formula for $Q$ , regardless of the order of $a(t)$ .

The effectivity argument on $R,T$ in Proposition 15.3 does not use $t\le2$ : it forces coefficients at least $(t-1)/2$ and $t-1$ whenever $t\ge1$ . Remove these components. Since $R,T$ are disjoint from $U$ , restriction to $U$ then has class $\mathcal O(2-t,0)$ . The normal bundle of $U$ forces a further coefficient at least $t-2$ for $t\ge2$ . After removing it, call the remaining class $M_t$ . We have $M_t|_U=\mathcal O(0,t-2),\qquad Q|_U=\mathcal O(1,1).$ The latter equality follows from $(H-2G-R-2T)\cdot Z=0$ before this last blow-up.

Let $l$ be the strict transform of a general line in $\Pi$ through $p_0$ . It meets $C$ only at $p_0$ , transversely, and its lift passes through $p$ . Its tangent at $p$ differs from those of $f_0$ and $Z$ . After the point blow-up it therefore avoids $f$ and meets the strict conic once, at their other plane intersection. Consequently $Hl=1,\quad Bl=2,\quad Gl=1,\quad Rl=Tl=0,\quad Ul=1,$ $Ql=-2,\qquad M_tl=5-2t.$ These general ruling curves cover $Q$ . The calculation concerns a general member and does not require $Q$ to be smooth everywhere.

Let $\gamma,\varepsilon$ be the coefficients of $Q,U$ in an effective representative of $M_t$ . Remove them. Restriction to $U$ , where the remaining divisor is effective, gives $\varepsilon\ge\gamma$ . Intersection with a general ruling of $Q$ gives $5-2t+2\gamma-\varepsilon\ge0.$ Hence $\gamma\ge2t-5$ and $\varepsilon\ge2t-5$ , proving (17.1). Notice that no bisecant curve has been blown up in this argument.

Apply this to a nonzero section of $mL$ vanishing to order at least $j$ at $p$ . If $j\ge5m/2$ , its divisor on $X$ contains the plane divisor, of class $H$ , with coefficient at least $2j-5m$ . But $mL-cH$ can be effective only if $c\le m$ : in the effective-cone basis $E,B$ its $E$ -coefficient is $(m-c)/3$ . Thus $2j-5m\le m$ , or $j\le3m$ . Finally, the plane section of $H$ has order two at $p$ , and a section of $B$ vanishing simply at $\varphi(p)$ pulls back to order one, since $\varphi$ is smooth there. Their product is a section of $L$ of order three. This proves the asserted threshold. ◻

Remark 17.2 (What was, and was not, decomposed). The two initial blow-ups give a genuine Zariski decomposition for $1\le t\le2$ . Beyond that interval, Lemma 17.1 gives necessary fixed components on a smooth modification. It does not assert that the remaining divisor is nef, or that all base curves have been resolved. Such assertions in higher contact would require additional work. The estimate below needs only inclusions of jet spaces, so these necessary components suffice. This is how we avoid a complete threefold Zariski decomposition in the higher-contact cases while still using the geometry of the blown-up base curves.

Corollary 17.3 (A further uniform improvement of the volume bound). For $p\in f_0$ on a smooth conic fibre, $S_X(G)\le\frac{9441}{4624},\qquad \frac3{S_X(G)}\ge\frac{4624}{3147}.$

Proof. Use the concavity estimate in Corollary 15.5, but now the volume vanishes for $t\ge3$ by Lemma 17.1. Thus $\int_2^3\mathop{\mathrm{vol}}(D_t)\,dt \le\frac{289}{56}\left(1-\left(\frac{10}{17}\right)^4\right) =\frac{10503}{2312}.$ Adding $\int_0^2\mathop{\mathrm{vol}}(D_t)\,dt=225/8$ and dividing by $16$ gives the result. ◻

Lemma 17.4 (Explicit spaces containing the jets). There are homogeneous coordinates $[u:v:w]$ on $G$ such that $q=[T_pf_0]=[0:0:1]$ , the infinitely near point $q_1$ has tangent direction $v=0$ , and the tangent conic of the plane surface is $K=(vw+c u^2=0)$ for some $c\in\mathbb C$ , possibly zero. For integers $m,j\ge0$ , put $k_{m,j}=\max\{0,2j-5m\},\quad d_{m,j}=j-2k_{m,j},\quad b_{m,j}=\max\{0,j-m-2k_{m,j}\}.$ For $0\le j\le3m$ , let $U_{m,j}$ be the span of the monomials $\label{f29:eq:monomial-envelope} u^i v^r w^{d_{m,j}-i-r},\qquad i,r\ge0,\quad i+r\le d_{m,j},\quad i+2r\ge b_{m,j}.$ Then $W_{m,j}\subset K^{k_{m,j}}U_{m,j}\subset H^0(G,\mathcal O_G(j)), \qquad W_{m,j}=0\quad(j>3m).$ Here $K^{k_{m,j}}$ means multiplication by the corresponding power of its equation.

Proof. Take $u=t,v=\eta,w=z$ in the tangent space of the local coordinates above. The quadratic initial form of $\eta z+a(t)$ is $vw+cu^2$ . Near $q$ , its equation has weighted order two for weights $(1,2)$ on $(u,v)$ . The second exceptional curve of the successive point blow-ups of $G$ computes exactly this weighted order. Indeed, in the chart $v=u v_1$ , the second blow-up at $u=v_1=0$ assigns orders one and two to $u,v$ .

The forced coefficient $j-m$ along $T$ gives weighted order at least $j-m$ for a degree- $j$ leading form when $j\ge m$ . Lemma 17.1 forces divisibility by $K^{k_{m,j}}$ . Dividing by it leaves degree $d_{m,j}$ and weighted order at least $j-m-2k_{m,j}$ . A polynomial has that weighted order precisely when its monomials satisfy (17.2). For $j\le m$ there is no additional condition. The preceding lemma gives the last vanishing assertion.

No completeness of a restricted positive part is used: the displayed relation is an inclusion. In particular, collisions among other tangent directions do not invalidate it. ◻

Lemma 17.5 (Comparing with monomial spaces). Let $V=\bigoplus_{m,j}V_{m,j}$ be a bigraded linear series on $G\cong\mathbb P^2$ , with bounded support and containing an ample series, in the sense of [4, Section 2.5]. These hypotheses hold for the refinement used here by [4, Example 2.15] and ensure the existence of the expected-vanishing limits. Suppose $V_{m,j}\subset K^{k_{m,j}}U_{m,j}$ , where $k_{m,j}\ge0$ , each $U_{m,j}$ is a space of homogeneous forms of a single degree spanned by monomials in $u,v,w$ , and $K=vw+cu^2$ . For each $m$ , assume that only finitely many $U_{m,j}$ are nonzero. Normalize basis divisors by $mN_m$ , where $N_m=\sum_j\dim V_{m,j}>0$ . Define $C_{u,m},C_{v,m},C_{w,m}$ by summing the respective exponents of all monomials of all $U_{m,j}$ and dividing by $mN_m$ . Put $C_{K,m}=\frac{\sum_j k_{m,j}\dim U_{m,j}}{mN_m},\qquad C_m=\max\{C_{u,m},C_{v,m}+C_{K,m},C_{w,m}+C_{K,m}\}.$ Then, for every divisorial valuation $\nu$ over $G$ , $S_m(V;\nu)\le C_m A_G(\nu).$ If the four constants have limits, the reciprocal of the limiting maximum is a lower bound for $\delta(G;V)$ , with the convention $1/0=+\infty$ .

Proof. All orders of zero divisors of nonzero sections are nonnegative. The dimension of each step of the valuation filtration of $K^{-k_{m,j}}V_{m,j}$ is at most the corresponding dimension for $U_{m,j}$ . Summing the dimensions of these steps bounds the maximal sum of valuation orders. Also $\dim V_{m,j}\le\dim U_{m,j}$ . It follows that $S_m(V;\nu)$ is at most $\nu(\mathcal D_m)$ , where $\mathcal D_m=C_{K,m}K+ \frac1{mN_m}\sum_{j,i}\operatorname{div}(s_{j,i})$ and each $\{s_{j,i}\}_i$ is a valuation-compatible basis of $U_{m,j}$ .

Choose an integer $M$ larger than every degree under consideration. The one-parameter action $[u:v:w]\longmapsto[\lambda^M u:\lambda v:w]$ has distinct weights on the monomials in each fixed degree. By the two-filtration lemma [4, Lemma 3.1], we can choose each basis simultaneously compatible with the valuation and this weight filtration. Its initial forms are then the distinct monomials spanning $U_{m,j}$ . The initial form of $K$ is $vw$ . Hence $\mathcal D_m$ degenerates to $C_{u,m}(u=0)+(C_{v,m}+C_{K,m})(v=0) +(C_{w,m}+C_{K,m})(w=0).$ If $C_m=0$ , all the sections involved are constant and the desired inequality is immediate. Otherwise, after division by $C_m$ , the limit is a simple-normal-crossing boundary with all coefficients at most one, and is therefore log canonical. Lower semicontinuity of log canonical thresholds [11, Theorem 0.2] shows that $(G,\mathcal D_m/C_m)$ is log canonical as well: all nonzero fibres of the degeneration are projectively equivalent. Thus $\nu(\mathcal D_m)\le C_m A_G(\nu)$ for every $\nu$ , proving the assertion. Passing to the expected-vanishing limits gives the last claim. This also proves that the comparison controls arbitrary divisors over the plane, not just monomial valuations. ◻

Lemma 17.6 (The four integrals for the comparison). For the spaces of Lemma 17.4, the constants in Lemma 17.5 converge to $C_u=\frac{387}{512},\qquad C_v=\frac{801}{1024},\qquad C_w=\frac{603}{1024},\qquad C_K=\frac{105}{1024}.$

Proof. Set $t=j/m$ and use the following piecewise linear functions: $\begin{array}{c|ccc} t&d(t)&a(t)&k(t)\\\hline {[0,1]}&t&0&0\\ {[1,5/2]}&t&(t-1)/2&0\\ {[5/2,3]}&10-3t&(9-3t)/2&2t-5. \end{array}$ The normalized exponent region is $\Omega_t=\{(x,y)\in\mathbb R_{\ge0}^2: x+y\le d(t),\ x+2y\ge2a(t)\}.$ In every row $d\ge2a\ge0$ . Thus $\Omega_t$ is the triangle of side $d$ with the corner triangle of vertices $(0,0),(2a,0),(0,a)$ removed. Its area and moments are $\begin{aligned} \int_{\Omega_t}1&=\frac{d^2}{2}-a^2,& \int_{\Omega_t}x&=\frac{d^3}{6}-\frac{2a^3}{3},\\ \int_{\Omega_t}y&=\frac{d^3}{6}-\frac{a^3}{3},& \int_{\Omega_t}(d-x-y)&=\frac{d^3}{6}-da^2+a^3. \end{aligned}$ For example, the removed triangle has area $a^2$ and centroid $(2a/3,a/3)$ , giving its first two moments immediately.

Since $N_m=(8/3)m^3+O(m^2)$ , lattice summation and division by $mN_m$ give the factor $3/8$ , not $3/16$ : $\begin{aligned} C_u&=\frac38\int_0^3\left(\frac{d^3}{6}-\frac{2a^3}{3}\right)dt,& C_v&=\frac38\int_0^3\left(\frac{d^3}{6}-\frac{a^3}{3}\right)dt,\\ C_w&=\frac38\int_0^3\left(\frac{d^3}{6}-da^2+a^3\right)dt,& C_K&=\frac38\int_{5/2}^3 k\left(\frac{d^2}{2}-a^2\right)dt. \end{aligned}$ The separate contributions, with the factor $3/8$ already included, are $\begin{array}{c|rrrr} t&C_u&C_v&C_w&C_K\\\hline {[0,1]}&1/64&1/64&1/64&0\\ {[1,5/2]}&1137/2048&2355/4096&1761/4096&0\\ {[5/2,3]}&379/2048&785/4096&587/4096&105/1024. \end{array}$ Adding the columns proves the formulas. Boundary lattice points and rounding contribute $O(m^3)$ to the numerator, hence vanish after division by $mN_m$ . The larger spaces have total asymptotic dimension $(73/64)N_m$ ; they have deliberately not been renormalized to their own dimension. ◻

Theorem 17.7 (All smooth conic fibres on the distinguished fibre). For every $p\in f_0$ whose conic fibre is smooth, $\delta(G;W)\ge\frac{512}{453},\qquad \delta_p(X)\ge\frac{512}{453}>1.$ No transversality assumption on $C\cap\Pi$ or $C\cap Z$ is required.

Proof. The three coefficients after the monomial degeneration are $C_u=\frac{387}{512},\qquad C_v+C_K=\frac{453}{512},\qquad C_w+C_K=\frac{177}{256}.$ Their maximum is $453/512$ . Lemmas 17.4–17.6 give the first inequality. By Lemma 14.1 and Corollary 17.3, $\delta_p(X)\ge \min\left\{\frac{4624}{3147},\frac{512}{453}\right\} =\frac{512}{453}.$ For the same conclusion the earlier, weaker bound $S_X(G)\le273/128$ would already suffice. Thus this conclusion does not depend on the complete transverse-case decomposition in Section 16. ◻

Example 17.8 (Why higher contact is allowed). If $a(t)=ct^2+O(t^3)$ with $c\ne0$ , the tangent conic $vw+cu^2=0$ is smooth. If $a(t)=ct^r+O(t^{r+1})$ with $r\ge3$ , its tangent conic is the reduced union $vw=0$ . In both cases its weighted order at $q$ is two and the one-parameter degeneration in Lemma 17.5 has initial form $vw$ . Thus the same numerical bound applies. These are local illustrations; the proof does not assume that the higher-contact case deforms with unchanged Zariski chambers.

Corollary 17.9 (A sharper bound in transverse position). Under (16.1), $\delta(G;W)\ge\frac{512}{431},\qquad \delta_p(X)\ge\frac{512}{431}.$

Proof. In the comparison lemma, the fixed-conic term may be normalized using the actual dimensions of $W_{m,j}$ ; only the moving part needs to be enlarged. Its limiting coefficient is $C_K^{\mathrm{actual}} =\frac1{16}\int_{5/2}^3(2t-5)(-V'(t))\,dt =\frac18\int_{5/2}^3V(t)\,dt =\frac{61}{1024},$ where $V(t)=\mathop{\mathrm{vol}}(D_t)$ is given by Proposition 16.3. Integration by parts explains the middle equality, and the last chamber integral is $61/128$ . The other three constants remain valid upper bounds. Thus the maximum coefficient becomes $\max\left\{\frac{387}{512},\frac{801+61}{1024}, \frac{603+61}{1024}\right\}=\frac{431}{512}.$ The same degeneration proves the asserted bound for $\delta(G;W)$ , and $3/S_X(G)=768/503>512/431$ proves the local bound. ◻

Corollary 17.10 (Reduction to double conic fibres). Every point on a reduced conic fibre satisfies $\delta_p(X)\ge96/95$ . If the conic bundle has no nonreduced fibre, then $\delta(X)\ge96/95$ , and $X$ is K-stable. In general, the centre of every divisor with nonpositive beta invariant is contained in the union of the reduced supports of the nonreduced conic fibres.

Proof. Combine Proposition 12.4 with the smooth-fibre estimate away from $f_0$ and the reduced-singular-fibre estimate in Section 12. Alternatively, Theorem 17.7 gives a second sufficient bound at the smooth-fibre points of $f_0$ . The least of the three bounds is $96/95$ . A centre containing a point with local threshold greater than one cannot have nonpositive beta invariant. If all fibres are reduced, this also gives the stated global delta bound. ◻

Lemma 17.11 (The different normal bundle at a double conic fibre). Suppose $\varphi^{-1}(y)=2\Lambda$ , where $\Lambda\cong\mathbb P^1$ is its reduced support. Then $L\cdot\Lambda=1,\qquad N_{\Lambda/X}\cong\mathcal O(1)\oplus\mathcal O(-2).$ For $p\in\Lambda$ , let $\Lambda_Y$ be its strict transform on $Y=\mathop{\mathrm{Bl}}_pX$ . Then $N_{\Lambda_Y/Y}\cong\mathcal O\oplus\mathcal O(-3).$ Blowing up $\Lambda_Y$ gives an exceptional surface $R_\Lambda\cong\mathbb F_3$ . If $e$ is its negative section, then $\bigl(\tau^*D_t-\alpha R_\Lambda\bigr)\cdot e=1-t \quad\text{for every real }\alpha.$ In particular, subtracting any multiple of this one exceptional divisor cannot produce a nef positive part for $t>1$ .

Proof. The reduced fibre is a line in the plane of the conic, so $H\Lambda=1$ and $B\Lambda=0$ . Locally on the base the conic bundle lies in a trivial plane bundle. Along $\Lambda$ , the normal sequence of its inclusion in that fourfold is the kernel of $\mathcal O(1)\oplus\mathcal O^{\oplus2} \longrightarrow\mathcal O(2).$ The map on $\mathcal O(1)$ is zero because the fibre equation is a square. The other two entries are binary quadrics $q_1,q_2$ with no common zero: a common zero would make the total threefold singular. Their kernel is $\mathcal O(-2)$ . Thus the displayed kernel is $\mathcal O(1)\oplus\mathcal O(-2)$ .

Blowing up a point of a smooth curve twists both normal summands by $\mathcal O(-1)$ , giving the second formula. On the exceptional $\mathbb F_3$ , with ruling class $r$ , the normal divisor is $-e-3r$ . The restriction of $\tau^*D_t$ is $(1-t)r$ . As $e^2=-3$ and $r\cdot e=1$ , the restriction of $\tau^*D_t-\alpha R_\Lambda$ has intersection $\bigl(\alpha e+(1-t+3\alpha)r\bigr)\cdot e=1-t.$ This proves the obstruction to stopping after this blow-up. It gives the starting geometry for the remaining calculation, not a completed decomposition there. ◻

Remark 17.12 (Changing the order of extraction). The smooth-fibre cases, including high contact at the distinguished point, are settled by the surface blow-up in Proposition 12.4 and independently by the exceptional-plane refinement. Lemma 17.11 explains why a double conic fibre requires different geometry. We now extract its reduced curve with weights $(1,2)$ before making any point extraction, and apply Abban–Zhuang to the resulting exceptional surface.