Section 3

Elementary estimates for refined series

Part I · Abban–Zhuang theoryPDF page 21

Lemma 3.1 (Complete series on a rational curve). Let a space of sections on $\mathbb P^1$ have a fixed effective rational divisor $F$ and complete moving part $H^0(\mathbb P^1,\mathcal O(\ell))$ . Its dimension is $\ell+1$ , and its mean order at a point $Q$ is $\operatorname{coeff}_QF+\ell/2$ .

Proof. The complete moving series has a basis of orders $0,1,\ldots,\ell$ at $Q$ . Restoring the fixed factor adds $\operatorname{coeff}_QF$ to every order. ◻

Lemma 3.2 (Toric alpha invariant). Let $T=\mathbb P(a_0,a_1,a_2)$ be well formed, $H_T=\mathcal O_T(1)$ , and $\Delta=\sum c_iD_i$ , where $D_i$ are the coordinate divisors and $0\le c_i<1$ . Then $\alpha(T,\Delta;H_T)=\min_i a_i(1-c_i)$ . For $T=\mathbb P(2,1,3)_{U,V,W}$ and $\Delta=\tfrac12(W=0)$ , the local alpha invariant of $H_T$ is at least $3/2$ on $V\ne0$ .

Proof. Take $D\ge0$ with $D\sim_{\mathbb Q}H_T$ and clear denominators. A generic one-parameter subgroup of the torus degenerates its defining polynomial to a monomial. The resulting divisor is $D_0=\sum d_iD_i$ , where $d_i\ge0$ and $\sum a_id_i=1$ . If $\lambda\le\min_i a_i(1-c_i)$ , then $c_i+\lambda d_i\le1$ for every $i$ . The toric pair $(T,\Delta+\lambda D_0)$ is log canonical, so lower semicontinuity of log canonical thresholds gives the same conclusion for $D$ . This proves the lower bound. Taking $D=D_i/a_i$ gives the reverse inequality at the generic point of $D_i$ .

On $V=1$ , sections of $kH_T$ are polynomials in $U,W$ of weighted degree at most $k$ , for weights $(2,3)$ . At a point of $W=0$ , translate $U$ to put the point at the origin. This preserves the degree bound and the boundary. Monomial degeneration there gives threshold at least $\min\{2,3/2\}/k$ . Away from $W=0$ , both coordinates may be translated and the same argument without boundary gives at least $2/k$ . This proves the local assertion. The semicontinuity argument applies to the germ at the fixed origin; translations do not increase weighted degree. ◻

Lemma 3.3 (Residual-degree estimate). Let $(G,\Delta_G)$ be a projective klt surface, let $H_G$ be ample and $\mathbb Q$ -Cartier, and suppose $\mathop{\mathrm{Cl}}(G)_{\mathbb Q}=\mathbb Q[H_G]$ . Let $B\sim_{\mathbb Q}b_0H_G$ be a prime curve. Assume that sections of $W_{m,j}$ have class $d_{m,j}H_G$ and that their orders along $B$ are integral. Set $h_{m,j,q}=\dim\mathop{\mathrm{gr}}_B^qW_{m,j}$ and suppose the limit $R=\lim_{m\to\infty}\frac1{mN_m}\sum_{j,q}h_{m,j,q}(d_{m,j}-b_0q)$ exists. If $\alpha(G,\Delta_G;H_G)\ge a>0$ , then $S(W;v)\le(R/a)A_{G,\Delta_G}(v)$ for every divisorial valuation whose centre is not contained in $B$ . The same conclusion holds for a chosen valuation if the local alpha bound $a$ holds at some point of its centre.

Proof. Choose a basis in every $W_{m,j}$ compatible with $v$ and $\mathop{\mathrm{ord}}_B$ , using Lemma 2.3. A basis section with order $q$ along $B$ has divisor $qB+D'$ , where $D'\ge0$ and $D'\sim_{\mathbb Q}(d_{m,j}-b_0q)H_G$ . Ampleness implies that the residual coefficient is nonnegative; if it is zero, $D'=0$ . Since the centre of $v$ is not contained in $B$ , one has $v(B)=0$ . The global or local alpha bound gives $v(s)=v(D')\le(d_{m,j}-b_0q)A_{G,\Delta_G}(v)/a$ . Summing and dividing by $mN_m$ proves the claim in the limit. Compatibility with $v$ computes the left side, while compatibility with $B$ fixes the multiplicities $h_{m,j,q}$ . ◻

Example 3.4. On $\mathbb P(1,1,2)$ with $\Delta=\frac12(V=0)+\frac23(W=0)$ , Lemma 3.2 gives alpha equal to $1/2$ . If $B\sim2H_G$ , a section of class $dH_G$ divisible by $B^q$ has residual class $(d-2q)H_G$ . Away from $B$ , its valuation is therefore at most $2(d-2q)A_{G,\Delta}(v)$ . Section 9 averages exactly these residual coefficients.