Section 6

Keeping information from more than one divisor

Part I · Abban–Zhuang theoryPDF page 30

The minimum in the adjunction inequality gives a convenient threshold bound, but it discards part of the discrepancy. The finite-degree inequality proved above retains that part. Two further forms used in this manuscript are recorded here.

6.1 Zeros away from the chosen point of a curve

Lemma 6.1. Let $W$ be a multigraded series on a normal projective surface $T$ , with bounded support and with all expected-order limits appearing below finite and existent. Suppose the numerical classes of its basis divisors tend to $D_W$ . Let $B,C$ be distinct rational Cartier prime curves and let $q\in C\setminus B$ . Assume restriction to $C$ defines the refinement under consideration, and $C$ is smooth near $q$ . Then $S(W^C;q)\le\bigl(D_W-S(W;B)B-S(W;C)C\bigr)\cdot C.$ The class inside the parentheses is a limit of effective classes. A known lower bound for $S(W;B)$ may replace it in the right side.

Proof. First take the associated graded for the $C$ -filtration. On each piece, the original subspaces with $B$ -order at least $j$ induce a filtration. Taking the $C$ -associated graded preserves their total dimensions, hence preserves their total $B$ -weight. Choose a basis of each piece compatible with this induced filtration and with the order at $q$ . Lift it with the indicated $C$ - and $B$ -orders. Once its $C$ factor is removed, a lift with $B$ -order $j$ still has the factor $(B|_C)^j$ , all of whose zeros are away from $q$ . The order at $q$ is at most the degree remaining after these zeros are removed. Sum and divide by $mN_m$ . The lifts also show that removal of both factors before restriction leaves an effective divisor. Passing to the limit proves both assertions. For the last statement, $B\cdot C\ge0$ , so decreasing the subtracted coefficient only weakens the upper bound. ◻

The argument uses two filtrations on each already restricted space; it does not assume that three arbitrary filtrations admit one compatible basis. This estimate is useful on the ruled exceptional surfaces in Family 2.14, where a large known mean on one section forces a smaller residual degree at the chosen point of a ruling; see Lemma 43.6.

6.2 Two effective adjunctions on a smooth model

Here is a general calculation which explains the smaller boundary appearing in Family 1.9. Work locally on a smooth birational model $\pi:Z\to X$ with smooth exceptional divisors $E,G$ , meeting transversely in $C$ . Suppose, in the neighbourhood being used, $K_Z=\pi^*K_X+(A_E-1)E+(A_G-1)G.$ For basis divisors adapted to the valuation being tested, write $\pi^*D_m=D'_m+b_mE+a_mG$ . Suppose that the actual refinement on $G$ has basis divisor $W_m=D'_m|_G+(b_m-\rho a_m)C, \qquad b_m\ge\rho a_m,$ where $\rho\ge0$ . For example, this occurs when the pullback of the weighted exceptional divisor to $Z$ is $G+\rho E$ .

Assume $\lambda_m>0$ is chosen so that $(G,\theta C+\lambda_mW_m)$ is log canonical near $q\in C$ , and $0\le\rho\lambda_m a_m-(A_E-1)\le\theta, \qquad \lambda_m b_m-(A_E-1)\ge0.$ After decreasing the coefficient of $C$ , adjunction and inversion of adjunction give log canonicity of the effective pair $\bigl(Z,G+\lambda_mD'_m+ (\lambda_m b_m-(A_E-1))E\bigr)$ near $q$ . Comparing its boundary with the pullback of $K_X+\lambda_mD_m$ yields $\label{az:eq:two-divisor-remainder} A_X(v)-\lambda_m v(D_m) \ge(A_G-\lambda_m a_m)v(G)$ for every valuation whose centre there is contained in $G$ . Indeed the restricted boundary is exactly $\lambda_mW_m+(\rho\lambda_m a_m-(A_E-1))C$ ; the only remaining difference of coefficients is on $G$ . Taking limits is legitimate for a basis adapted to $v$ , because $v(D_m)=S_m(v)$ . A finite-degree threshold slightly below its limit supplies the required $\lambda_m$ ; one must not replace asymptotic log canonicity by finite-degree log canonicity without this step.

Example 6.2 (The special line in Family 1.9). In Lemma ?? the smooth model has $A_E=2$ , $A_G=3$ , $\rho=1/2$ and $a_m\longrightarrow\frac{43}{18},\qquad \frac{a_m}{2}\le b_m\le S_m(E),\qquad S_m(E)\longrightarrow\frac{101}{72}.$ The surface calculation proves $\delta_q(G,7C/36;W)\ge1$ . Notice that $(43/18)/2-1=7/36$ . Taking $\lambda_m\to1$ with $\lambda_ma_m\le43/18$ makes the preceding boundaries effective for large $m$ and gives $A_X(v)-S_X(v)\ge \left(3-\frac{43}{18}\right)v(G) =\frac{11}{18}v(G)>0$ when the centre is contained in $G$ . The usual different of the weighted extraction is still $C/2$ . The coefficient $7/36$ is an effective boundary on this smooth two-divisor model obtained by the displayed calculation, not a replacement value for that different.

Sometimes $\lambda_m b_m-(A_E-1)$ is negative. Ordinary inversion of adjunction for effective boundaries does not justify keeping that coefficient. In that situation we use its nonnegative part and verify the new restricted coefficient. The local identity in Lemma ?? makes this explicit. There $F'$ and $E$ are smooth and transverse, $W'_m$ is effective and does not contain $J$ , and the identities are $K_Z=\pi^*K_X+4F'+E,\quad \pi^*D_m=D'_m+a_mF'+e_mE,\quad D'_m|_{F'}=W'_m+(d_m-e_m)J,$ where $J=E|_{F'}$ and $d_m\ge e_m$ . Use the effective boundary $B_m=\lambda D'_m+(\lambda e_m-1)_+E$ . Its restriction is $B_m|_{F'}=\lambda W'_m+ (\lambda d_m-\min\{\lambda e_m,1\})J.$ If the last coefficient belongs to $[0,1]$ and $\lambda W'_m|_J$ has every coefficient at most one near the centre in question, inversion of adjunction on $F'$ and then on $Z$ gives $A_X(v)-\lambda v(D_m)\ge (5-\lambda a_m)v(F')+(1-\lambda e_m)_+v(E).$ Both inversions have effective boundary. The numerical input for that application is $a_m\to287/72$ , $d_m\to115/72$ , $e_m\ge(a_m-1)/4$ , and point orders on $J$ at most $1/3+o(1)$ . For $1<\lambda<288/245$ these verify the stated coefficient conditions. Indeed, when $\lambda e_m\le1$ the coefficient is at most $\lambda(d_m-(a_m-1)/4)\to245\lambda/288<1$ ; otherwise it tends to $115\lambda/72-1<1$ . The limiting coefficient on $F'$ at $\lambda=288/245$ is $5-(288/245)(287/72)=11/35>0$ . The local restriction identity and its section-theoretic origin are proved in the cited lemma.

Example 6.3 (A refined threshold equal to one). For one of the ordinary point configurations of Family 2.12, the refined series on $G=\mathbb P^2$ has a monomial limit with mean exponents $1,53/80,47/80$ . Its surface threshold is at least one; the first mean prevents that calculation from giving a larger bound. Nevertheless $A_X(G)=3$ and $S_X(G)=9/4$ . The finite-degree adjunction inequality retains $A_X(v)-S_X(v)\ge\left(3-\frac94\right)v(G) =\frac34v(G)>0$ for every divisor centred at the extracted point. The actual initial algebra and the moment calculation are proved in Lemma 35.4. A surface bound equal to one is therefore compatible with strict K-stability on the threefold; the positive remaining discrepancy is part of the proof.