Refinement by the exceptional divisor of the original blow-up

Proof. On $E=\mathbb P^1\times\mathbb P^1$, one has $D_u=2g+(1+u)s$. Each summand is complete on rational interior rays. For either of the two coordinate fibres of class $g$, the expected coefficient is $$\frac16\int_0^1 2^2(1+u)du=1;$$ for either coordinate fibre of class $s$, it is $$\frac16\int_0^1 2(1+u)^2du=\frac79.$$ A common monomial degeneration of the complete bidegree series, as in Lemma 5.3, proves that its threshold is at least one. The divisor of class $g$ proves equality. There is no different since both $X$ and $E$ are smooth. The strict adjunction inequality of Lemma 21.2, with $S_X(E)=5/9$, gives the stated beta inequality. ◻

Proof. Write $a=1+u$ and $b=3+u=a+2$. For subtraction of $s$, the divisor $(a-v)s+bg$ is nef for $0\le v\le a$, which is its threshold. Thus $$S(W^E;s)=\frac16\int_0^1\left(ba^2-\frac23a^3\right)du=\frac{71}{72}.$$ For subtraction of $g$, the divisor $as+(b-v)g$ is nef up to $v=b-2a$. Thereafter its positive part is $(b-v)(s+2g)/2$, and the negative part is $(v-b+2a)s/2$. The threshold is $b$. Hence $$S(W^E;g)=\frac16\int_0^1\left[ \int_0^{b-2a}(2ab-2a^2-2av)dv+ \int_{b-2a}^{b}\frac{(b-v)^2}{2}dv\right]du=\frac{29}{24}.$$ The other torus-invariant section has expected coefficient $41/72$, and the other invariant fibre has coefficient $29/24$. For example these numbers follow by summing distances to the four edges of the section polygon; the coefficients of the two section divisors sum to $\frac16\int_0^1aD_u^2du=14/9$. The toric degeneration lemma therefore gives the lower bound $24/29$, and a ruling fibre gives equality. This is the threshold of the actual asymptotically complete refined series, not an overestimate obtained by enlarging it. ◻