Weighted extraction of a double conic fibre

Proof. Locally on the base the conic bundle is a quadratic hypersurface in a trivial plane bundle. Let $x$ be a local equation of the line in its plane, and let $u,v$ be parameters at $y$. Its equation, to the relevant weighted order, is $$x^2+u q_1(s_0,s_1)+v q_2(s_0,s_1) +\text{terms of weighted order greater than two}=0,$$ where the weights of $x,u,v$ are $1,2,2$. The binary quadrics $q_1,q_2$ have no common zero, since otherwise $X$ would be singular at such a point of $\Lambda$. Where $q_1$ is a unit we can eliminate $u$ and write $\mathcal J=(x^2,v)$. Thus the extraction has transverse weights $(1,2)$.

The two charts of the blow-up of $(x^2,v)$ are the smooth chart $v=x^2r$ and the chart $x^2=vr$. The latter is an $A_1$ surface singularity times the parameter along $\Lambda$, hence is normal. The exceptional Cartier divisor has multiplicity two on its reduced support.

For a global description, use flatness of $\varphi$ from Lemma 10.5. Formation of the Rees algebra, and hence of the blow-up, commutes with flat base change. Thus $$X^\sharp=X\times_{\mathbb P^2}\mathop{\mathrm{Bl}}_y\mathbb P^2.$$ The exceptional scheme is $(2\Lambda)\times\mathbb P^1$, whose reduction is $F=\Lambda\times\mathbb P^1$, with second coordinate $[u:v]$. The pullback of the exceptional divisor of the blown-up base is $2F$. Restricting its normal bundle to $F$ gives $2F|_F=-f$. The singular-locus curve of $X^\sharp$ on $F$ is $$D=(q_1(s_0,s_1)u+q_2(s_0,s_1)v=0)\subset F.$$ It is the graph of a degree-two morphism $\Lambda\to\mathbb P^1$, and is therefore smooth of bidegree $(2,1)$.

On the smooth chart, $dx\wedge dv=x^2dx\wedge dr$, giving $K_{X^\sharp}=\omega^*K_X+2F$ and $A_X(F)=3$. On the other chart use the double cover $x=ab$, $v=a^2$, $r=b^2$, with $(a,b)\mapsto(-a,-b)$. This is an index cover for $F$, which has Cartier index two along the singular locus; the threefold $X^\sharp$ itself is Gorenstein there. The divisor $F$ is the image of $a=0$. The quotient is free in codimension one, the pair with this divisor is plt, and adjunction gives different $\tfrac12D$. Here “plt” means purely log terminal; in particular, the exceptional divisor is normal and its adjoint pair is klt. Also $-F$ is relatively ample, since $-2F$ is the tautological divisor of the blow-up. This verifies the hypotheses for Abban–Zhuang refinement.

Finally, $L\Lambda=1$ gives $\omega^*L|_F=h$. The adjunction identity can also be checked numerically: $$-h+3(-f/2)=K_F+\tfrac12D=-2h-2f+\tfrac12(2h+f).$$ ◻

Proof. The pencil of lines through $y$ resolves on $X^\sharp$, so $\omega^*B-2F$ is globally generated. Both $\omega^*H$ and $\omega^*B$ are nef, which proves nefness on the stated interval. On $F$ the intersection formulas of Lemma 18.1 give $$(\omega^*L)^2F=0,\qquad \omega^*L\cdot F^2=-\tfrac12, \qquad F^3=0.$$ Taking the cube proves the formula. Its contribution to $S_X(F)$ is $\frac1{16}\int_0^2(16-3t^2/2)dt=7/4$; this is not the whole expected vanishing order. ◻

Proof. Choose base coordinates with $y=[1:0:0]$. In the graph description of $X$, the first equation is $l_0y_0+l_1y_1+l_2y_2=0$, and $\Pi=(l_0=0)$. The order of $l_0$ along $F$ is two, with initial form $$l_1|_\Lambda\,u+l_2|_\Lambda\,v.$$ The two linear forms do not both vanish identically on $\Lambda$, because the common zero of $l_0,l_1,l_2$ is only $p_0$. This proves both class formulas. The curve $R$ may be reducible; no smoothness assumption on it will be made.

Take a general line $g\subset\Pi$ through $p_0$, and let $\widetilde g$ be its lift. It meets $C$ only at $p_0$, transversely, so $H\widetilde g=1$ and $B\widetilde g=2$. On $g$, the ratio $l_1:l_2$ is constant. The graph equation therefore shows that $\varphi(\widetilde g)$ is a line through $y$, and the map to that line has degree two. Thus the pullback of $\mathfrak m_y$ to $\widetilde g\cong\mathbb P^1$ has degree two. It is invertible, so the curve lifts to $X^\sharp$, with $2F\widetilde g=2$. This argument applies whether or not $\Lambda$ passes through $p_0$.

Consequently, $Q\widetilde g=-1$ and $D_t^\sharp\widetilde g=3-t$. These curves cover $Q$. Removing its coefficient $c$ from any effective representative gives $3-t+c\ge0$, hence $c\ge t-3$.

For a section of $mL$ of order at least $j$ along $F$, this forces a plane component of coefficient at least $j-3m$. As before, $mL-cH$ is effective only if $c\le m$, by the effective cone in Lemma 11.1. Hence $j\le4m$. Conversely, the plane section of $H$ has order two, as does a general section of $B$ through $y$. Their product has order four, proving the threshold assertion. ◻

Proof. Define the valuation ideals $\mathcal I_j=\omega_*\mathcal O_{X^\sharp}(-jF)$, where the sheaves on $X^\sharp$ are divisorial sheaves. The weighted normal algebra has generators $x,u,v$ of weights $1,2,2$ and initial relation $x^2+q_1u+q_2v=0$. Eliminating $x^2$ gives the following sheaves on $\Lambda$: $$\begin{aligned} \mathcal I_{2r}/\mathcal I_{2r+1} &\cong\mathcal O_\Lambda\otimes\operatorname{Sym}^r\mathbb C^2,\\ \mathcal I_{2r+1}/\mathcal I_{2r+2} &\cong\mathcal O_\Lambda(-1)\otimes\operatorname{Sym}^r\mathbb C^2. \end{aligned}$$ These descriptions glue globally. The two base parameters are global generators on a neighbourhood of $y$. The initial term of a change of the weight-one coordinate is multiplied by the transition function of the conormal line $\mathcal O_\Lambda(-1)$; all terms involving base parameters have higher weight. Thus the even part is generated by the base parameters, while the odd part has the additional conormal factor.

Restriction of sections gives an injection $$W^F_{m,j}\hookrightarrow H^0\bigl(\Lambda,L^m|_\Lambda\otimes (\mathcal I_j/\mathcal I_{j+1})\bigr).$$ Since $L|_\Lambda=\mathcal O_\Lambda(1)$, the target is identified with $H^0(F,\mathcal O_F(m,r))$ for $j=2r$, and with $H^0(F,\mathcal O_F(m-1,r))$ for $j=2r+1$. In the odd case the effective rational zero divisor also contains $D/2$: the square of the odd factor has equation $q_1u+q_2v$, whose divisor is $D$. This is exactly the fractional term prescribed by reflexive restriction.

This is also the reflexive restriction convention of [4, Lemma 2.7 and Example 2.15]. It is essential to retain the half divisor for odd weights, rather than treating $F$ as Cartier. Lemma 18.4 forces $k_{m,j}$ copies of $Q$; restricting and removing them gives exactly (18.1). Only an inclusion is claimed. Extra conditions caused by intersections with exceptional curves are harmless for the upper bounds below. ◻

Proof. For a complete series of bidegree $(A,B)$, the maximal sum of orders along a fixed fibre of the first projection is $A(A+1)(B+1)/2$, and that along a fibre of the second projection is $B(A+1)(B+1)/2$. These are the sums of exponents of a monomial basis. For any curve of bidegree $(2,1)$, the maximal sum of orders is $$\sum_{r\ge1}h^0(F,\mathcal O_F(A-2r,B-r)).$$ Since $N_m=(8/3)m^3+O(m^2)$, summing over $j$ and taking limits gives exactly the displayed integrals. The $O(1)$ changes in bidegrees for odd $j$ do not affect the limits. The separate contributions are $$\begin{array}{c|ccccc} t&c_h&c_f&c_R&c_D&c_F\\\hline {[0,1]}&3/64&1/64&0&3/256&1/16\\ {[1,3]}&3/8&13/32&0&5/32&13/8\\ {[3,4]}&11/128&43/256&5/64&5/128&53/64. \end{array}$$ For example, the upper limit in the $c_D$ integral is $t/2$ on $[0,1]$, $1/2$ on $[1,3]$, and $(4-t)/2$ on $[3,4]$. Thus every integral is a polynomial integral on the listed intervals.

For the last assertion, Lemma 18.5 gives $\dim W^F_{m,j}\le\dim U_{m,j}$. Therefore $$S_m(F)=\frac{\sum_j j\dim W^F_{m,j}}{mN_m} \le\frac{\sum_j j\dim U_{m,j}}{mN_m}\longrightarrow c_F.$$ The total dimension of the larger spaces has leading ratio $\frac38\int_0^4ab\,dt=35/32$ to $N_m$. We continue to normalize by $N_m$, not by the dimension of these larger spaces. ◻

Proof. The projection $D\to\mathbb P^1_{[u_0:u_1]}$ has degree two. Its two ramification points and two branch points are distinct in characteristic zero. Moving them to zero and infinity gives the stated equation. The invariant-curve list follows either from the invariant function $s_0^2u_1/(s_1^2u_0)$ or by factoring a weight-homogeneous bihomogeneous polynomial.

Outside $o_0,o_\infty$, the listed curves have only simple normal crossings, so the coefficient condition suffices. At $o_0$, use coordinates $(s,u)$. The curves through the point are $s=0$, $u=0$, and $u=s^2/c$. Let their coefficients be $a,b,c_1,\ldots,c_r$. The two successive ordinary point blow-ups resolve this collection. The exceptional valuations have weights $(1,1)$ and $(1,2)$ and log discrepancies two and three. The second required inequality is $$a+2b+2\sum_i c_i\le3.$$ Together with $a\le1$, it implies $a+b+\sum_i c_i\le2$, which is the inequality for the first blow-up. After the second blow-up, the parabolas meet the exceptional curve at distinct points; all remaining intersections are simple normal crossings. This proves log canonicity at $o_0$. The coordinates at $o_\infty$ give the identical calculation. ◻

Proof. We apply the surface point-extraction inequality to the actual series $W^F$, with boundary $D/2$. This is the form of Abban–Zhuang refinement explained in [3, Corollary 1.102 and Remark 1.113], with the boundary retained as in [4, Theorem 3.3]. For a plt extraction $\sigma:\widetilde F\to F$ at $p$, with exceptional curve $T$, write $V^T$ for the refinement of $W^F$ by $T$ and $\Delta_T$ for the different of $(\widetilde F,T+\frac12\widetilde D)$. Then $$\delta_p(F,\tfrac12D;W^F)\ge \min\left\{\frac{A_{F,\frac12D}(T)}{S(W^F;T)}, \inf_{q\in T}\frac{1-\operatorname{coeff}_q\Delta_T}{S(V^T;q)}\right\}.$$ All moments below are normalized by $mN_m$, where $N_m=h^0(X,mL)$, even when we enlarge the refined spaces to $U_{m,j}$. We first explain this comparison for a point extraction. Removing the factors in (18.1) gives an inclusion into $U_{m,j}$. After shifting the exceptional order by $k_{m,j}\operatorname{ord}_T(R)+(\epsilon_j/2)\operatorname{ord}_T(D)$, its associated graded spaces inject into those of $U_{m,j}$. A basis in each such space can be extended to a basis of the larger space. All orders of effective zero divisors, including their effective restrictions to $T$, are nonnegative. Thus the maximal sums of orders are bounded above by those for the complete spaces. The fixed factor $k_{m,j}R$ contributes its order along $T$ to the first moment and the restriction of its strict transform to the point moments on $T$. Replacing the dimensions of the actual spaces by $\dim U_{m,j}$ bounds both contributions from above. The additional factor $\epsilon_jD/2$ contributes $O(m^{-1})$ to all the moments considered below. This follows because $\epsilon_j\le1$, while the sum of the dimensions of the $U_{m,j}$ is $O(m^3)$ and $mN_m$ has order $m^4$.

Fix $p\in F$ and choose affine coordinates $(s,u)$ from the two factors, both zero at $p$. A basis of $H^0(F,\mathcal O_F(A,B))$ is given locally by $s^iu^j$, with $0\le i\le A$ and $0\le j\le B$. First let $\sigma$ be the ordinary blow-up of $p$. The order of this monomial along $T$ is $i+j$. The two ruling directions give points $q_s,q_u\in T$, and its restriction after removing this exceptional order has orders $i,j$ at these points. Therefore the complete-space contributions to the three moments are respectively $c_h+c_f$, $c_h$, and $c_f$.

At any other point $q\in T$, the degree-$k$ initial space has dimension $$n_{A,B}(k)=\max\{0,\min(A,k)-\max(0,k-B)+1\}.$$ After removing its common zeros at $q_s,q_u$, it is the complete space of polynomials of degree $n_{A,B}(k)-1$ on $T\cong\mathbb P^1$. Its maximal sum of orders at $q$ is $n_{A,B}(k)(n_{A,B}(k)-1)/2$. If $a,b$ denote the functions in Lemma 18.6 and $$d(t,v)=\max\{0,\min(a(t),v)-\max(0,v-b(t))\},$$ then the normalized limiting sum is $$c_{\mathrm{pt}}=\frac38\int_0^4\int_0^{a(t)+b(t)}\frac{d(t,v)^2}{2}\,dv\,dt =\frac38\int_0^4\left(\frac{\min(a,b)^2\max(a,b)}2 -\frac{\min(a,b)^3}6\right)dt =\frac{43}{128}.$$ For completeness, the contributions to the last integral before multiplication by $3/8$ are $1/4$, $11/24$, and $3/16$ on $[0,2]$, $[2,3]$, and $[3,4]$, respectively.

A local equation of $R$ is $\alpha+\beta s+\gamma u+\eta su$. Hence $\operatorname{mult}_pR\le2$. If this multiplicity is two, the initial equation is a nonzero multiple of $su$, so its two tangent directions are distinct. If the multiplicity is one, its initial equation is linear. In either case the restriction of the strict transform of $R$ to $T$ has coefficient at most one at every point. This also covers reducible $R$: a divisor of bidegree $(1,1)$ is either irreducible or the reduced union of one fibre of each ruling. Consequently $$\begin{split} S(W^F;T)&\le c_h+c_f+2c_R=\frac{321}{256},\\ S(V^T;q_s)&\le c_h+c_R=\frac{75}{128},\qquad S(V^T;q_u)\le c_f+c_R=\frac{171}{256},\\ S(V^T;q)&\le c_{\mathrm{pt}}+c_R=\frac{53}{128} \quad(q\ne q_s,q_u). \end{split}$$ These estimates are uniform in $p$ and $q$: the finite-dimensional formulas depend only on $A,B$, and the fixed-factor multiplicities are uniformly bounded as above.

If $p\notin D$, the different on $T$ is zero and $A_{F,\frac12D}(T)=2$. The point-extraction inequality gives $\delta_p(F,\frac12D;W^F)\ge256/171$. Suppose next that $p\in D$ is not a ramification point of the degree-two projection $D\to\mathbb P^1_{[u_0:u_1]}$. Since the other projection restricts to an isomorphism on $D$, the tangent direction of $D$ at $p$ is neither ruling direction. Thus $\Delta_T=\frac12q_D$, with $q_D\ne q_s,q_u$, and $A_{F,\frac12D}(T)=3/2$. It follows that $$\delta_p(F,\tfrac12D;W^F)\ge \min\left\{\frac{128}{107},\frac{128}{75}, \frac{256}{171},\frac{64}{53}\right\} =\frac{128}{107}>\frac{128}{123}.$$

It remains to consider the two ramification points. Choose product coordinates there so that $D=(u=s^2)$. Blow up $p$ and then blow up the intersection of the first exceptional curve with the strict transform of $D$. Contracting the strict transform of the first exceptional curve, which has self-intersection $-2$, gives the weighted blow-up with weights $\operatorname{ord}_T(s)=1$ and $\operatorname{ord}_T(u)=2$. Its exceptional curve is $T=\mathbb P(1,2)$. The weighted blow-up has a quotient singularity of type $\frac12(1,1)$ at $q_s=(s=0)$ and is smooth elsewhere on $T$. The strict transform of $D$ meets $T$ transversally at the smooth point $q_D=(u/s^2=1)$, which differs from $q_s$ and $q_u=(u=0)$. Adjunction and the discrepancy formula therefore give $$\Delta_T=\tfrac12q_s+\tfrac12q_D,\qquad A_{F,\frac12D}(T)=1+2-\tfrac12\cdot2=2.$$ In particular, $(\widetilde F,T+\frac12\widetilde D)$ is plt, so the same point-extraction inequality applies.

For the complete bidegree spaces the exceptional order is now $i+2j$, giving the moment $c_h+2c_f$. The orders on $T$ at $q_s,q_u$ are $i/2,j$, respectively; the factor $1/2$ is the quotient-index contribution at $q_s$. At any other point, say $u/s^2=c\ne0$, order at least $r$ is equivalent to divisibility of the weighted initial polynomial by $(u-cs^2)^r$. Summing over all weighted degrees, the dimension of these divisible polynomials is $$(A-2r+1)_+(B-r+1)_+, \qquad x_+:=\max\{x,0\}.$$ Thus the maximal normalized sum of orders at this point is $c_D=53/256$, by exactly the dimension sum in Lemma 18.6.

The four monomials $1,s,u,su$ in the equation of $R$ have distinct weights $0,1,2,3$. Its initial equation is therefore one monomial $s^iu^j$ with $i,j\in\{0,1\}$. Its exceptional order is at most three, and the restriction of its strict transform to $T$ has coefficients at most $1/2$ at $q_s$ and one at $q_u$, with no other points in its support. Consequently (18.2)$$\label{f29:eq:weighted-branch-bound} \begin{split} S(W^F;T)&\le c_h+2c_f+3c_R=\frac{123}{64},\\ S(V^T;q_s)&\le\frac{c_h+c_R}{2}=\frac{75}{256},\qquad S(V^T;q_u)\le c_f+c_R=\frac{171}{256},\\ S(V^T;q)&\le c_D=\frac{53}{256}\quad(q\ne q_s,q_u). \end{split}$$ The point-extraction inequality now yields $$\delta_p(F,\tfrac12D;W^F)\ge \min\left\{\frac{128}{123},\frac{128}{75}, \frac{256}{171},\frac{128}{53}\right\} =\frac{128}{123}.$$ Every point of $F$ belongs to one of the three cases. Taking the infimum proves the asserted global threshold bound, including valuations whose centres on $F$ are curves. ◻

Proof. The extraction in Lemma 18.1 is of plt type, with centre $\Lambda$ and discrepancy three. Abban–Zhuang refinement applies with the different $\tfrac12D$. The local thresholds of the refined surface along centres mapping to $p$ are bounded below by its global threshold. Thus $$\delta_p(X)\ge \min\left\{\frac3{S_X(F)},\delta(F,\tfrac12D;W^F)\right\} \ge\min\left\{\frac{192}{161},\frac{128}{123}\right\} =\frac{128}{123}.$$ The refined series has bounded support by Lemma 18.4; it contains an ample series, as in [4, Example 2.15]. Concretely, $\omega^*L-tF$ is ample for sufficiently small positive $t$, since $L$ is ample and $-F$ is relatively ample. This also verifies the positivity hypothesis needed for refinement. ◻

Proof of Theorem 1.1 for Family 2.9. A conic fibre is smooth, a reduced union of two lines, or a double line. Sections 12 and 17 give $\delta_p(X)\ge96/95$ at every point of a reduced fibre. Theorem 18.10 gives the stronger bound $128/123$ at every point of a double fibre. Therefore $\delta(X)\ge96/95>1$, proving K-stability for every smooth member of Family 2.9. ◻