Fibre refinements and reduction of centres

Proof. The nef cone is generated by $H,F$, and the effective cone by $E,F$. For the latter assertion, a prime divisor other than $E$ is the proper transform of a divisor of class $aH$ on $V_5$, of multiplicity $b$ along $C$, where $a\ge1$ and $b\ge0$. Restricting it to a general fibre gives $a\ge b$, and its class is $aF+(a-b)E$. For $0\le u\le1$, the divisor $L-uF=H+(1-u)F$ is nef. For $1\le u\le2$, its divisorial decomposition is $$P(u)=(2-u)H,\qquad N(u)=(u-1)E.$$ Indeed, intersection with a general fibre of $E\to C$ forces the displayed multiplicity along $E$, and the remainder is nef. The effective cone gives pseudoeffective threshold two. Therefore $$S_X(F)=\frac1{20}\left(\int_0^1(20-15u)\,du+ \int_1^2 5(2-u)^3\,du\right)=\frac{11}{16}.$$ The divisors $L-uE=(1-u)H+(1+u)F$ and $L-uH=(1-u)H+F$ are nef for $0\le u\le1$; both have pseudoeffective threshold one. The intersection numbers give $$S_X(E)=\frac14\int_0^1(4-6u+2u^3)\,du=\frac38,\qquad S_X(H)=\frac14\int_0^1(4-9u+6u^2-u^3)\,du=\frac5{16}.$$ Finally, $P(u)|_F=-K_F$ for $u\le1$ and $P(u)|_F=(2-u)(-K_F)$ for $u\ge1$, while $N(u)|_F=(u-1)C$ on the latter interval. The moving and fixed coefficients in the restriction formula are respectively $$\frac{15}{20}\left(\int_0^1 1\,du+\int_1^2(2-u)^3\,du\right) =\frac{15}{16},\qquad \frac{15}{20}\int_1^2(2-u)^2(u-1)\,du=\frac1{16}.$$ This proves the exact refinement formula; it is also the degree-five instance of the computation in [8, Lemma 2.1]. These decompositions occur on $X$ itself. No decomposition on a threefold obtained by extracting a point or a line is used in this lemma. ◻

Proof. Use Lemma 44.1 and Abban–Zhuang along $F$. At a point off $E$, every valuation entering the local estimate has zero order on $C$. At a point of $C$, the inequality $\operatorname{ord}_T(C)\le A_F(T)$ follows from log canonicity of $(F,C)$. For a smooth quintic del Pezzo surface, $\delta(F)=15/13$ by [3, Lemma 2.11]. ◻

Proof. Choose a basis of $H^0(X,-mK_X)$ simultaneously adapted to $Q$ and $F$. Write its basis divisor as $D_m=a_mF+\Gamma_m$, where $a_m=S_m(F)$ and $F\not\subset\operatorname{Supp}(\Gamma_m)$. Its restriction $\Gamma_m|_F$ is a basis divisor of the refined series. Convergence of the finite basis thresholds gives numbers $c_m\to1$, with $0<c_m\le1$, such that $(F,c_m\Gamma_m|_F)$ is log canonical near $p$. Inversion of adjunction gives $$A_X(Q)\ge\operatorname{ord}_Q(F)+c_m\operatorname{ord}_Q(\Gamma_m) =c_m S_m(Q)+(1-c_ma_m)\operatorname{ord}_Q(F).$$ Taking the limit and using $S_X(F)=11/16$ proves the assertion. The last inequality is strict because the centre of $Q$ is contained in $F$. Only prime divisors over $X$ are being used; this is sufficient for the valuative criterion in this paper. ◻

Proof. The exact refinement by a fibre is $S(W^F;v)=\frac{15}{16}S_F(v)+\frac1{16}v(C)$. Since $F$ and $C$ are smooth along $C$, the pair $(F,C)$ is log canonical. Thus $\delta_p(F)\ge1$ implies $S(W^F;v)\le A_F(v)$ for every prime divisor $v$ over $F$ with centre containing $p$. Lemma 44.3 applies.

In the two remaining cases, put $L=L_1$ or $L'_1$ as appropriate. The curve $C$ meets $L$ once and transversally: $C\sim-K_F$, $(-K_F)\cdot L=1$, and $F$ is smooth along $C$. For type $A_2$, the positive part of $-K_{\widetilde F}-v\widetilde L$ has square $5-2v-v^2/3$ for $0\le v\le3/2$ and $(2-v)(4-v)$ for $3/2\le v\le2$. Its intersection with $\widetilde L$ is respectively $1+v/3$ and $3-v$. Consequently $$S_F(L)=\frac{31}{30},\qquad S(W_F^L;p)=\frac15\left(\int_0^{3/2}(1+v/3)^2\,dv+ \int_{3/2}^2(3-v)^2\,dv\right)=\frac{19}{30}.$$ For type $A_3$, the corresponding square and intersection are $(2-v)(10+v)/4$ and $1+v/4$, for $0\le v\le2$, giving $$S_F(L)=\frac{16}{15},\qquad S(W_F^L;p)=\frac15\int_0^2(1+v/4)^2\,dv=\frac{19}{30}.$$ These are the decompositions in [12, Lemmas 5.4–5.5, pp. 16–17]. The denominator in the displayed square in Lemma 5.5 is corrected here from $2$ to $4$; the stated $S$-value in that lemma is unchanged. The negative parts meet $\widetilde L$ only above the singular point of $F$, so they contribute nothing at $p$.

Refining $W^F$ by $L$, its first invariant is $31/32$ in type $A_2$ and $1$ in type $A_3$. The secondary invariant at $p$ is in both cases $$\frac{15}{16}\cdot\frac{19}{30}+\frac1{16}=\frac{21}{32}.$$ Here the last term is the restriction of the fixed boundary to the reduced point $C\cap L$. The different is zero near $p$. Abban–Zhuang on $F$ therefore gives refined threshold at least $32/31$ and $1$, respectively. Apply Lemma 44.3 once more. ◻

Proof. If the centre is a surface $D$, then $Q=\operatorname{ord}_D$. The cases $D=E$ and $D=F$ are excluded by $S_X(E)=3/8$ and $S_X(F)=11/16$. Every other prime surface is the proper transform of a divisor in $|aH|$ on $V_5$, of multiplicity $b$ along $C$, with $a\ge1$ and $0\le b\le a$. If $a=b$, the surface is a fibre, since $|aF|$ is pulled back from the base. Otherwise $D\sim aF+(a-b)E$, so $D-E$ is linearly equivalent to an effective divisor and $S_X(D)\le S_X(E)<1$.

On a smooth fibre $\delta(F)=15/13$. The exact fibre refinement gives $\delta_p(X)\ge16/13$ off $E$ and $\delta_p(X)\ge8/7$ on $E$. Thus any centre meeting a smooth fibre is excluded. In particular, every curve centre is contained in a singular fibre.

At a smooth point of a singular fibre off $E$, the local threshold table [12, Tables 10–15, p. 26] gives $\delta_p(F)>15/16$, except on the listed lines and on $L_1$ of a fibre of type $A_3$. In the latter case $\delta_p(F)=15/16$, so the refined threshold is at least one and Lemma 44.3 excludes every centre contained in $F$. At points on $E$, the same tables and Lemma 44.4 exclude all points outside the listed lines. For a curve centre we apply these arguments at a general closed point of the curve.

Finally, the incidence configurations in [12, Figure 33, p. 25] show that every singular point of a fibre of type $2A_1$, $A_2$, $A_2+A_1$, $A_3$ or $A_4$ lies on a listed line. This leaves precisely the centres in the statement. ◻