Anticanonical sections and the distinguished surface

Proof. The anticanonical class is $h+f_1+H$, while the hypersurface has class $h+2H$. Its restriction sequence has kernel $$\mathcal O_{\mathbb F_1\times\mathbb P^2} \bigl((m-1)h+mf_1+(m-2)H\bigr).$$ For $m=1$, the factor $\mathcal O_{\mathbb P^2}(-1)$ has no cohomology. For $m\ge2$, pushing the first factor down to $\mathbb P^1$ gives line bundles of degrees $2m-1,2m-2,\ldots,m$, all nonnegative. Künneth’s formula therefore gives zero first cohomology in either case, proving surjectivity. Comparing the two degrees on $\mathbb F_1$ gives precisely the displayed monomials.

The dimensions of the ambient space and the kernel are $$\frac{(m+1)^2(3m+2)(m+2)}4, \qquad \frac{m^2(3m+1)(m-1)}4,$$ respectively. Their difference is the asserted dimension. This proof uses the homogeneous coordinates of the ambient product and its hypersurface equation; it requires no assertion about generators of the whole Cox ring of $X_U$. ◻

Proof. For $0\le u\le3/2$, the decomposition of $-K_X-uE$ is $$\begin{array}{c|c|c} u&P(u)&N(u)\\ \hline {[0,1]}&(1-u)h+H+(1+u)f_1&0\\ {[1,3/2]}&(3-2u)H+2f_1&(u-1)E'. \end{array}$$ The positive parts are nef, since $h,H,f_1$ are pullbacks of globally generated divisors. The second row is the fixed-part decomposition: on a general exceptional fibre of $\pi$, the original divisor has degree $1-u$, so every effective representative contains each component of $E'$ with coefficient at least $u-1$. After this part is removed, the remainder is pulled back from $\mathbb P^1\times\mathbb P^2$ and is nef. The same argument applies to the nodal member, whose centre is a reduced union of two smooth components. Their exceptional divisors are both generically ordinary blow-ups, so the coefficient bound applies to each. For $u>3/2$ the residual plane degree $3-2u$ is negative, proving that the threshold is $3/2$. The common intersection numbers give $$S_X(E)=\frac1{24}\left( \int_0^1(24-12u-6u^2)\,du+ \int_1^{3/2}6(3-2u)^2\,du\right)=\frac{17}{24}.$$

Write $P(u)|_E=a(u)\alpha+b(u)\beta$, where $$(a(u),b(u))= \begin{cases} (1+u,2),&0\le u\le1,\\ (2,6-4u),&1\le u\le3/2. \end{cases}$$ For $0\le v\le b(u)$ the divisor $P(u)|_E-vC=a(u)\alpha+(b(u)-v)\beta$ is nef; it has volume $2a(u)(b(u)-v)$ and intersection $a(u)$ with $C$. Since $C$ is not a component of $E'|_E$, there is no primary fixed contribution. Therefore $$\begin{aligned} S(W^E;C) &=\frac18\int_0^{3/2}\int_0^{b(u)} 2a(u)(b(u)-v)\,dv\,du\\ &=\frac18\left(\int_0^1 4(1+u)\,du+ \int_1^{3/2}2(6-4u)^2\,du\right)=\frac{11}{12}. \end{aligned}$$ The moving point contribution is $$\frac18\int_0^{3/2}a(u)^2b(u)\,du =\frac18\left(\int_0^1 2(1+u)^2\,du+ \int_1^{3/2}4(6-4u)\,du\right)=\frac56.$$ The curve $E'|_E$ meets $C$ transversally at one point. For a smooth member this is the graph property. For the nodal member its other component is disjoint from $C$ by assumption, and its graph component has the same property. The only possible fixed point contribution is consequently $$\frac14\int_1^{3/2}\int_0^{6-4u}2(u-1)\,dv\,du =\frac1{24}.$$ This proves the three claimed values and bounds. The nodal singularity is disjoint from $E$, so the same adjunction inequality applies there. Finally, $$\delta_p(X)\ge \min\left\{\frac{24}{17},\frac{12}{11},\frac87\right\} =\frac{12}{11}.$$ ◻

Proof. This is the local refinement inequality of [4, Theorem 3.3]. We recall its basis-divisor argument. Choose an anticanonical basis compatible with the order along $G$ and with a valuation being tested. Removing the powers of a local equation of $G$ gives bases of the spaces $W^G_{m,j}$. The pullback basis divisor has the form $$S_m(G)G+B'_m,$$ where $B'_m$ does not contain $G$ and its restriction is a basis divisor of $W^G$ with the usual normalization by $m h^0(X,mL)$. If a positive number $c$ is smaller than both quantities on the right, the proof of [4, Lemma 2.21] gives $$S_m(W^G;v)\le(1+\epsilon)S(W^G;v)$$ for all valuations of finite discrepancy and all sufficiently large $m$, with the bound on $m$ independent of $v$. Thus the same strict inequalities hold for the basis thresholds simultaneously over the proper fibre $\pi^{-1}(p)\cap G$. Then adjunction makes $(G,\Delta_G+cB'_m|_G)$ log canonical over $p$. Inversion of adjunction applies to $(Y,G+cB'_m)$ there. The coefficient of $G$ in the pullback of the corresponding pair on $X$ is $$1-A_X(G)+cS_m(G)<1.$$ Decreasing the coefficient of $G$ therefore gives log canonicity of the original basis pair at $p$. Taking the infimum over the valuations and the limit of the basis thresholds proves the inequality. Equivalently, this is the same refinement argument as for a point extraction, with the whole fibre over $p$ retained. ◻