The weighted extraction of a double fibre

Proof. In a local trivialization of the plane bundle, the equation has the form $x^2+\eta_1Q_1(s_0,s_1)+\eta_2Q_2(s_0,s_1)+$ higher weighted-order terms, with weights $1,2,2$. The quadrics $Q_1,Q_2$ have no common zero, by smoothness of $X$. Locally the base ideal is $(x^2,\eta_2)$. Its blow-up has charts $\eta_2=x^2r$ and $x^2=\eta_2r$; the latter has an $A_1$ singularity transverse to the singular-locus curve. Flat base change from $\mathop{\mathrm{Bl}}_y\mathbb P^2$ shows that the reduced exceptional surface is the product, and that the pullback of the exceptional base curve is $2T$. The singular-locus curve is $D=(\eta_1Q_1+\eta_2Q_2=0)$, giving the different $D/2$. The Jacobian on the first chart gives discrepancy three. Finally $L\Lambda=1$, since $L$ has degree two on the conic fibre. This proves all the formulas, including the case $\Lambda\subset E$. ◻

Proof. On the plane over $y$, the rank-one quadratic is the square of a linear form in $u,v,ex_0$. If its line is different from $e=0$, a linear change in $u,v$, followed by addition of a multiple of $ex_0$, makes this form $u$. If its line is $e=0$, normalize its nonzero coefficient to one. Taking initial terms gives the two equations. The two binary quadrics are the first base derivatives on $\Lambda$, so they have no common zero by smoothness of the total space.

For completeness, the section ring calculation is global. On the plane bundle $Y\to\mathbb P^2$, the pushforward of $\mathcal O_Y(pH+qB)$ for $p\ge0$ is $$\bigoplus_{a=0}^p\mathcal O_{\mathbb P^2}(q+a)^{\oplus(p-a+1)}.$$ Thus $H^1(Y,(m-2)H+(m-1)B)=0$. Restriction from $Y$ to the hypersurface of class $2H+B$ identifies $H^0(X,mL)$ with the bidegree $(m,m)$ quotient by $G$.

The weighted hypersurface filtration has associated graded quotient by $G_0$. This is the one-equation case of Lemma 5.1. Each displayed $G_0$ is a primitive linear polynomial in $x_1,x_2$ over the ring of the other variables and is irreducible; the coprimality of the binary quadrics ensures primitivity. Hence this quotient is a domain, and it injects into its localization at $x_0$ and a nonzero homogeneous coordinate on $\Lambda$. On that chart the weighted normal equation is precisely the local equation in Lemma 27.1. Its associated graded ring is the geometric one. The two inverted weight-zero coordinates have bidegrees forming a lattice basis. In any fixed bidegree, their common Laurent factor is therefore uniquely determined, and removing it preserves injectivity. The injectivity criterion in Lemma 5.1 now proves equality of the global quotient and geometric filtrations. In particular, no unaccounted cancellation is used to assign a section its order. ◻

Proof. We give a separate Hilbert calculation. For the polynomial Cox ring, let $P(p,q)$ be the dimension of degree $pH+qB$, with $p,q\ge0$. A monomial has $e$-exponent $a$, total $u,v$-degree $p-a$, and total $x$-degree $q+a$. Therefore $$P(p,q)=\sum_{a=0}^p(p-a+1)\binom{q+a+2}{2}.$$ For $d=q+a$, the sum of the $x_1,x_2$ weights over all degree-$d$ base monomials is $2d(d+1)(d+2)/3$. In the first case, the additional weight sum of the $u,v$ monomials is $(p-a)(p-a+1)/2$. In the second case, each has the additional weight $a$. Consequently the polynomial weight sums are $$\begin{split} U(p,q)&=\sum_{a=0}^p\left[\frac{(p-a)(p-a+1)}2\binom{d+2}{2} +(p-a+1)\frac{2d(d+1)(d+2)}3\right],\\ V(p,q)&=\sum_{a=0}^p(p-a+1)\left[a\binom{d+2}{2} +\frac{2d(d+1)(d+2)}3\right]. \end{split}$$ The relation has bidegree $(2,1)$ and weight two. Subtract its multiples, including their additional weight two. The quotient dimension is $$N_m=P(m,m)-P(m-2,m-1)=\frac{(2m+1)(3m^2+3m+2)}2.$$ The two quotient weight sums are, respectively, $$\begin{split} U(m,m)-U(m-2,m-1)-2P(m-2,m-1) &=\frac{m(169m^3+230m^2+191m+58)}{24},\\ V(m,m)-V(m-2,m-1)-2P(m-2,m-1) &=\frac{m(91m^3+128m^2+107m+34)}{12}. \end{split}$$ Divide by $mN_m$ and take the limit. Lemma 27.2 identifies these algebraic weights with the geometric valuation, so Lemma 2.1 applies. ◻