The refined series on the weighted exceptional surface

Proof. In the first equation of Lemma 27.2, put $x_0=1$. On $\Lambda$, the restrictions of $v,e$ give independent linear coordinates, with $e=r$. A monomial with $e$-exponent $a$ and $u$-exponent $\epsilon+2h$ has remaining $v$-degree $M-a-2h$ and base weight degree $J-h$. Its existence before localization is exactly the condition $$a,h\ge0,\quad M-a-2h\ge0,\quad J-h\ge0, \quad J-h\le m+a.$$ Using $u^2=-Q$ on this chart gives the forms $r^a v^{M-a-2h}Q^h$ times arbitrary binary base forms of degree $J-h$. The final inequality is $a+h\ge J-m$. Their span is precisely the indicated graded component of $(r,Q)^n$. Conversely every generator of that component satisfies these inequalities and lifts to a Cox monomial, using the nonnegative remaining power of $x_0$. Thus the assertion is an equality of spaces. An odd power of $u$ contributes the rational factor $D/2$, since its square restricts to $Q$. ◻

Proof. The map whose graph is $D$ has degree two. Put its ramification and branch points at zero and infinity. The action $([s_0:s_1],[t_0:t_1])\mapsto([\lambda s_0:s_1],[\lambda^2t_0:t_1])$ preserves $D$, and, after interchanging zero and infinity if needed, has the flat limit $R=(s_0=0)$. This degeneration of ideal components is flat, not merely an inclusion of limiting generators: $r_\lambda,Q$ are a regular sequence for all parameters, including the limit. The powers of their ideal have generators of degrees $(n+h,h)$ for $0\le h\le n$, and the consecutive syzygies have degrees $(n+h+2,h+1)$ for $0\le h<n$. The resulting two-term free resolution gives a constant Hilbert function in every bidegree. Thus the generator limit has the full required dimension. Finally $(s_0,Q)=(s_0,s_1^2t_0)$.

A monomial $s_0^A s_1^{M-A}t_0^C t_1^{J-C}$ lies in this ideal power exactly when for some $k$ one has $A\ge n-k$, $M-A\ge2k$ and $C\ge k$. Equivalently, $$A\ge\max\{0,2n-M\},\qquad C\ge\max\{0,n-A\}.$$ Divide by $m$. These are (28.1), up to $O(1/m)$ changes of the walls. The region is empty for $t>4$. ◻

Proof. Let $\Omega$ be the region in (28.1) and define $\langle\phi\rangle=\frac13\int_\Omega\phi(t,a,b)\,dt\,da\,db$. The factor $1/3$ comes from $N_m\sim3m^3$. To make these polynomial integrations explicit, split the region into $$\begin{array}{c|c|c} t&a&b\\\hline {[0,2]}&[0,1]&[0,t/2]\\ {[2,3]}&[0,t/2-1]&[t/2-1-a,t/2]\\ {[2,3]}&[t/2-1,1]&[0,t/2]\\ {[3,4]}&[t-3,t/2-1]&[t/2-1-a,t/2]\\ {[3,4]}&[t/2-1,1]&[0,t/2]. \end{array}$$ Direct integration gives $$\langle1\rangle=1,\quad \langle t\rangle=169/72,\quad \langle a\rangle=5/9,\quad \langle1-a\rangle=4/9, \quad\langle b\rangle=29/48,\quad\langle t/2-b\rangle=41/72.$$ These are the four divisor coefficients. The two weight sums are $\langle a+2b\rangle=127/72$ and $\langle1-a+2(t/2-b)\rangle=19/12$. For instance, on the first row $\langle a\rangle=1/6$ and $\langle b\rangle=1/9$; the other four rows contribute $7/18$ and $71/144$, respectively.

To bound the order along $D_c$, take a monomial order in which its initial term is $s_1^2t_0$. Choose a $D_c$-adapted basis and put its initial forms in echelon form for this monomial order. Their pivot monomials are distinct. If a form in one of the limiting monomial spaces is divisible by $D_c^k$, its initial term is divisible by $s_1^{2k}t_0^k$ and still belongs to that monomial space. Thus the sum of orders along $D_c$ is bounded by one half of the sum of the $s_1$ exponents, with a bounded rounding error per section. Normalization gives at most $(4/9)/2=2/9$. ◻

Proof. Use the second equation of Lemma 27.2. If the exponent of $e$ is $\epsilon+2h$, the remaining $u,v$-degree is $M-2h$ and the base weight degree is $J-h$. The unused $x_0$-exponent is nonnegative precisely when $J-h\le m+\epsilon+2h$. The least possible $h$ is therefore $h_{m,j}$. For a fixed $h$ all forms in the indicated bidegree occur. The subspaces for larger $h$ are contained in the one for the least $h$, by multiplication by $Q$. This proves equality and the degree table.

The four required integrals, retaining normalization $mN_m$, are $$c_h=\frac16\int a^2b\,dt=\frac38,\qquad c_f=\frac16\int ab^2\,dt=\frac{41}{72},\qquad c_D=\frac13\int dab\,dt=\frac18,$$ $$c_{\mathrm{mov},D}=\frac13\int\!\int_0^{\min(a/2,b)}(a-2z)(b-z)\,dz\,dt=\frac{11}{72}.$$ For the last integral the upper limit is $t/2$ on $[0,1]$, $1/2$ on $[1,2]$, and $(5-t)/6$ on $[2,5]$. The contributions on these three intervals are respectively $1/96$, $7/144$, and $3/32$. For the first three integrals, the contributions from $[0,2]$ and $[2,5]$ are, respectively, $$(1/6,\,1/9,\,0),\qquad (5/24,\,11/24,\,1/8).$$ These formulas follow by summing exponents in complete bidegree spaces and by summing dimensions after subtracting $kD$, as in Lemma 18.6. In addition $\frac13\int ab\,dt=1$ and $\frac13\int tab\,dt=91/36$, agreeing with the independent Hilbert calculation. ◻

Proof. We use the invariant-boundary criterion of Lemma 18.7. In the coordinates above, an invariant effective divisor is supported on the four coordinate fibres and finitely many curves $D_c$. It is log canonical if all prime coefficients are at most one and its weights at the two ramification points are at most three. This criterion is proved by two ordinary point blow-ups at each ramification point; the different contributes weight one there.

Fix a divisorial valuation on $T$. In each refined summand choose bases compatible with that valuation and with the filtration of the action preserving $D$, using Lemma 2.3. In the first case, Lemma 28.2 identifies their initial spaces exactly. Their initial bases are weight-homogeneous bases of those spaces, though individual forms need not be monomials. Their coordinate coefficients are bounded by the maxima in Lemma 28.3. Their $D_c$ coefficients are at most $2/9+o(1)$. Their two weight sums are $127/72+o(1)$ and $19/12+o(1)$: the sum of weights of a weight-compatible basis is independent of that basis. The parity factor contributes only $O(1/m)$.

For any fixed $0<\lambda<144/127$, all prime coefficients of the limiting pair with boundary $D/2+\lambda$ times the normalized initial basis divisor are less than one for large $m$. Indeed the largest coordinate coefficient is at most $\lambda\,29/48<1$, and that of $D$ is at most $1/2+\lambda\,2/9<1$. At the two ramification points the weight is at most $$1+\lambda\,127/72+o(1)<3.$$ The invariant-boundary criterion proves log canonicity of the limit. Semicontinuity, with $D$ preserved throughout the degeneration, gives log canonicity of the original pair.

In the second case use the complete moving spaces and fixed factors from Lemma 28.4. The coordinate coefficients are at most $3/8$ and $41/72$, and the coefficient of $D$ is at most $1/8+11/72=5/18$. At either ramification point the expected weight, including the fixed factor, is $$c_h+2c_f+2c_D=\frac38+\frac{82}{72}+\frac14=\frac{127}{72}.$$ The same argument applies, since $\lambda\,41/72<1$ and $1/2+\lambda\,5/18<1$. Compatibility with the original valuation converts log canonicity into its discrepancy bound. Taking limits and then letting $\lambda$ increase to $144/127$ proves the proposition for every divisorial valuation. ◻

Proof. The plt extraction of Lemma 27.1 has discrepancy three and different $D/2$. Abban–Zhuang refinement along its centre gives $$\delta_p(X)\ge\min\{3/S_X(T),\delta(T,D/2;W)\}.$$ The two possible first entries are $216/169$ and $108/91$, both larger than $144/127$. Bounded support follows from the explicit refined series (with thresholds four and five). The ample-series hypothesis holds since $L$ is ample and $-T$ is relatively ample; hence $\omega^*L-tT$ is ample for all sufficiently small positive rational $t$. Thus the refinement theorem applies. ◻