Section 2

Stability thresholds and refinement

Part I · Abban–Zhuang theoryPDF page 13

Abban–Zhuang theory

This part collects the definitions, adjunction inequalities and computational tools used throughout the manuscript. We state the hypotheses on the actual section spaces, explain the normalization of their expected vanishing orders, and retain fixed divisors and differents in every restriction. The general refinement theorem is due to Abban and Zhuang [4]; its surface applications are developed in [3, Section 1.7], and the rational-divisor formulation used here is given in [17]. The examples below are taken from the later proofs, where their geometric inputs and full calculations are established.

2.1 Discrepancies, alpha and delta

Let $Y$ be a normal projective variety and let $\Delta$ be an effective rational divisor such that $K_Y+\Delta$ is $\mathbb Q$ -Cartier. A prime divisor over $Y$ is a prime divisor $F$ on a normal variety $Y'$ with a projective birational morphism $f:Y'\to Y$ . Its centre is $c_Y(F)=f(F)$ , and its log discrepancy is $A_{Y,\Delta}(F)=1+\operatorname{coeff}_F\bigl(K_{Y'}-f^*(K_Y+\Delta)\bigr).$ One may compute this on a resolution; it is independent of the chosen model. The pair is log canonical, respectively klt, if every such discrepancy is nonnegative, respectively positive. We write $A_Y(F)$ when $\Delta=0$ . A divisorial valuation is a positive multiple of $\mathop{\mathrm{ord}}_F$ ; discrepancy and all expected vanishing orders are homogeneous in that multiple.

For an effective $\mathbb Q$ -Cartier rational divisor $D$ on a klt pair, its log canonical threshold is $\mathop{\mathrm{lct}}(Y,\Delta;D)=\sup\{c\ge0:(Y,\Delta+cD)\text{ is log canonical}\} =\inf_{F}\frac{A_{Y,\Delta}(F)}{\mathop{\mathrm{ord}}_F(D)},$ where a zero denominator is interpreted as $+\infty$ . If $M$ is an ample rational divisor, we define $\alpha(Y,\Delta;M)=\inf_{D\ge0,\ D\sim_{\mathbb Q}M}\mathop{\mathrm{lct}}(Y,\Delta;D).$ The local alpha invariant is defined by requiring log canonicity at the specified point. This is the algebraic alpha invariant; for the anticanonical polarization it corresponds to Tian’s alpha invariant [6]. For $a>0$ , the inequality $\alpha(Y,\Delta;M)\ge a$ means that $v(D)\le A_{Y,\Delta}(v)/a$ for every such $D$ and every divisorial valuation $v$ . When $Y$ is Fano, $\alpha(Y)$ means $\alpha(Y,0;-K_Y)$ .

For a big rational Cartier divisor $M$ on an $n$ -fold, define its volume by $\mathop{\mathrm{vol}}(M)=\limsup_{m\to\infty}\frac{n!h^0(Y,mM)}{m^n},$ where $m$ is divisible enough to make $mM$ integral. Volume extends continuously to real divisor classes. Choose a smooth projective model $f:Y'\to Y$ carrying a given prime divisor $F$ , so the volume expressions below involve Cartier real divisors. Set $S_M(F)=\frac1{\mathop{\mathrm{vol}}(M)}\int_0^\infty\mathop{\mathrm{vol}}(f^*M-tF)\,dt, \qquad \delta(Y,\Delta;M)=\inf_F\frac{A_{Y,\Delta}(F)}{S_M(F)}.$ For an irreducible subvariety $Z\subset Y$ , the local invariant $\delta_Z(Y,\Delta;M)$ restricts the infimum to $F$ with $Z\subset c_Y(F)$ . In particular, $\delta_P(Y,\Delta;M)$ tests every centre containing $P$ , including curves and divisors through $P$ . It does not test only divisors whose centre equals $P$ . The corresponding local log canonical threshold requires log canonicity at the generic point of $Z$ . For a smooth Fano threefold, write $S_X(F)=S_{-K_X}(F)$ , $\delta(X)=\delta(X,0;-K_X)$ and $\delta_P(X)=\delta_P(X,0;-K_X)$ .

We use K-stability through its equivalent valuative formulation: $X$ is K-stable if $\beta_X(F)=A_X(F)-S_X(F)>0$ for every prime divisor $F$ over $X$ . K-semistability is characterized by the corresponding non-strict inequalities. The inequality $\delta(X)>1$ characterizes uniform K-stability. For a smooth Fano variety, the equivalence between K-stability and uniform K-stability follows from [21]; in particular K-stability is equivalent to $\delta(X)>1$ in our setting. These characterizations, and their relation to the definition using test configurations, are supplied by [18, 22, 6, 21]; they are external inputs here.

Lemma 2.1 (Expected vanishing from an adapted basis). Let $M$ be ample and Cartier and put $N_m=h^0(Y,mM)$ . Filter $H^0(Y,mM)$ by the subspaces $\mathcal F^j=\{s:\mathop{\mathrm{ord}}_F(s)\ge j\}$ . If $\mathcal B_m$ is a compatible basis, then $S_M(F)=\lim_{m\to\infty}\frac1{mN_m}\sum_{s\in\mathcal B_m}\mathop{\mathrm{ord}}_F(s).$

Proof. Compatibility means that each filtered subspace is spanned by the basis vectors it contains. Therefore the sum of the basis orders equals $\sum_{j\ge1}\dim\mathcal F^jH^0(Y,mM)$ . On a model carrying $F$ , the summand is $h^0(Y',mf^*M-jF)$ . Divide by $mN_m$ and regard the sum as a Riemann sum with $t=j/m$ . The asymptotic formula for sections, together with the linear bound on the filtration and the standard convergence theorem for filtered series [6], gives the volume integral above. The denominator uses $N_m=\mathop{\mathrm{vol}}(M)m^n/n!+O(m^{n-1})$ . ◻

Example 2.2. For $Y=\mathbb P^1$ , $M=\mathcal O_{\mathbb P^1}(d)$ and $F=Q$ a point, an adapted basis of $H^0(Y,mM)$ has orders $0,1,\ldots,md$ . Its average order divided by $m$ is $d/2$ . Equivalently, $S_M(Q)=d^{-1}\int_0^d(d-t)\,dt=d/2$ . Thus $\delta(\mathbb P^1)=1$ when $M=-K_{\mathbb P^1}$ .

2.2 The section spaces and their normalization

The auxiliary spaces in an adjunction argument are usually not complete linear systems. We specify them before discussing their invariants. Fix rational Cartier divisors $L_0,\ldots,L_r$ on a normal projective variety $T$ . A multigraded linear series $V$ consists of finite-dimensional subspaces $V_{m,\mathbf a}\subset H^0(T,mL_0+a_1L_1+\cdots+a_rL_r), \qquad (m,\mathbf a)\in\mathbb N\times\mathbb N^r,$ with $V_{0,\mathbf0}=\mathbb C$ and products contained in the space whose index is the sum of the indices. A divisor representing each $L_i$ is fixed. Thus, if $s$ is represented by a rational function in the indicated space, its effective zero divisor is $mL_0+\sum a_iL_i+\operatorname{div}(s)$ . For a rational divisor $D$ , the notation $H^0(T,D)$ means $H^0(T,\mathcal O_T(\lfloor D\rfloor))$ , but its section divisors retain the fractional part of $D$ .

The support is the closed convex cone generated by indices of nonzero spaces. We require its slice $m=1$ to be bounded. We say that $V$ contains an ample series if the support has nonempty interior, every integral index in its interior has nonzero sufficiently large multiples, and there is an interior integral index $\boldsymbol\alpha$ for which the associated divisor is $A+E$ , with $A$ ample rational Cartier and $E$ effective rational Cartier, such that multiplication by the section of $kE$ embeds $H^0(T,kA)$ in $V_{k\boldsymbol\alpha}$ for all sufficiently divisible $k$ . These are the hypotheses on series used in this part. We work in the lattice generated by the grading; when necessary a finite-index Cartier sublattice is chosen explicitly.

For a fixed first degree $m$ , set $N_m(V)=\sum_{\mathbf a}\dim V_{m,\mathbf a}$ . Bounded support makes the sum finite. An $m$ -basis divisor is $D_m=\frac{1}{mN_m(V)} \sum_{\mathbf a}\sum_{s\in\mathcal B_{m,\mathbf a}}\operatorname{div}_0(s),$ where $\mathcal B_{m,\mathbf a}$ is a basis of $V_{m,\mathbf a}$ . The denominator is the dimension of the whole degree- $m$ series, multiplied by $m$ . It is not the dimension of one summand. For a divisorial valuation $v$ , define $S_m(V;v)=\sup_{D_m}v(D_m)$ , and define $\delta_m(T,\Delta;V)$ as the infimum of $\mathop{\mathrm{lct}}(T,\Delta;D_m)$ over these divisors. The local version $\delta_{Z,m}$ uses the generic point of the irreducible subvariety $Z$ . Zero-dimensional summands contribute nothing.

The filtration of a summand by $v$ is $\mathcal F_v^\lambda V_{m,\mathbf a}=\{s:v(\operatorname{div}_0(s))\ge\lambda\}$ . It has finitely many jumps. A compatible basis is a basis for which every filtered subspace is spanned by a subset. Its orders maximize the sum of orders among all bases. Thus it realizes the supremum in $S_m(V;v)$ . This is the same linear-algebra argument used in Lemma 2.1.

Lemma 2.3 (Two compatible filtrations). Two finite, exhaustive and separated decreasing filtrations of a finite-dimensional vector space admit a common compatible basis.

Proof. Number the distinct steps of the filtrations $F^i$ and $G^j$ . In each intersection $F^i\cap G^j$ , choose a complement to $(F^{i+1}\cap G^j)+(F^i\cap G^{j+1})$ , and choose a basis of that complement. Descending induction shows that the chosen vectors with indices at least $(i,j)$ span $F^i\cap G^j$ . The dimension identity for a sum of two subspaces shows, by the same induction, that their number equals the dimension of the intersection. They are therefore independent. Taking the first intersection gives the required basis. This also proves the version for any filtrations with finitely many distinct steps; see [4, Lemma 3.1]. ◻

In particular, when estimating one chosen valuation, we may require its basis to be compatible also with a divisor used for restriction. More generally, a flag defines one lexicographically ordered filtration. Apply the lemma to that single filtration and the valuation filtration. No assertion that three arbitrary filtrations admit a simultaneous compatible basis is needed.

Proposition 2.4 (Passage to the limit). Let $(T,\Delta)$ be klt, and let $V$ have bounded support and contain an ample series. Then $S(V;v)=\lim_m S_m(V;v)$ exists for every divisorial valuation $v$ . Moreover, $\lim_m\delta_{Z,m}(T,\Delta;V) =\delta_Z(T,\Delta;V) =\inf_{Z\subset c_T(v)}\frac{A_{T,\Delta}(v)}{S(V;v)}.$ The same assertion without the centre condition gives the global invariant. The infimum can be taken either over prime divisors or over nontrivial real valuations of finite log discrepancy. For every $\epsilon>0$ there is $m_0$ , independent of $v$ , such that $S_m(V;v)\le(1+\epsilon)S(V;v)\qquad(m\ge m_0)$ for all valuations of finite log discrepancy and all degrees in which the series is nonzero.

Proof. The convergence and its uniform upper estimate are the multigraded convergence theorem [4, Lemma 2.21 and its proof]. Its hypotheses apply to the actual series $V$ , not just to the divisor classes containing its summands. For completeness, the implication for thresholds follows directly from the uniform estimate. At finite degree, exchanging the two infima over basis divisors and valuations gives $\delta_{Z,m}=\inf_{Z\subset c_T(v)}A_{T,\Delta}(v)/S_m(V;v)$ . The uniform estimate gives $\delta_{Z,m}\ge\delta_Z/(1+\epsilon)$ . Conversely, fixing any one divisorial valuation in this infimum and using convergence of its expected order gives $\limsup_m\delta_{Z,m}\le A_{T,\Delta}(v)/S(V;v)$ . Take the infimum and then let $\epsilon$ decrease to zero. The global argument is identical. ◻

If $V'_{m,\mathbf a}=V_{km,k\mathbf a}$ , with every grading coordinate multiplied by the same positive integer $k$ , then $S(V';v)=kS(V;v),\qquad \delta_Z(T,\Delta;V')=\frac1k\delta_Z(T,\Delta;V).$ This is the Cartier Veronese comparison [4, Lemma 2.24]. It permits one to establish a rational-divisor statement on a Cartier sublattice and then return to the original normalization. Selecting only some residue classes while retaining the original denominator does not give this comparison. In explicit Hilbert sums below we keep all nonzero summands; alternatively both numerator and denominator must be changed according to a specified Veronese series.

Lemma 2.5 (Comparison of filtered moments). For each $m$ , let $U_m\subset V_m$ be finite-dimensional vector spaces with the filtration induced by the same order function. Suppose their weights lie in $[0,Cm]$ , with $C$ independent of $m$ . Let $w(U_m),w(V_m)$ be the sums of the weights of compatible bases, and let $d_m=\dim V_m-\dim U_m$ . Then $0\le w(V_m)-w(U_m)\le Cm d_m$ . The same assertion holds after summing over the degree- $m$ summands. If $\dim V_m$ has order $m^n$ and $d_m=o(m^n)$ , the normalized moments differ by $o(1)$ , also when each is normalized by its own dimension. The estimate persists after an associated graded construction provided the subsequent filtration on the smaller space is induced from the larger one.

Proof. Equip $V_m/U_m$ with its quotient filtration. At every step the induced, ambient and quotient filtered pieces form an exact sequence. Their weight sums are additive, so $w(V_m)-w(U_m)=w(V_m/U_m)$ is the sum of $d_m$ numbers in $[0,Cm]$ . Division by $m\dim V_m$ proves the assertion with a common denominator. Changing the denominator to $m\dim U_m$ changes the normalized moment by at most $C d_m/\dim V_m$ . Associated graded dimensions preserve the deficit in the exact sequence, so the same argument applies to any subsequent induced filtration. Uniform bounds on $C$ and $d_m$ give a uniform error in any additional parameter, such as a point on a fixed curve. ◻

2.3 Actual restriction and the different

Let $F$ be a prime divisor on a normal model $\pi:Y\to T$ . Suppose either that $\pi$ is the identity, or that its only exceptional divisor is $F$ , with $F$ rational Cartier and $-F$ relatively ample. Starting with Cartier grading divisors, put $D_{m,\mathbf a}=\pi^*(mL_0+\sum a_iL_i)$ and define the refinement by $\begin{aligned} W_{m,\mathbf a,j} &=\frac{\mathcal F_F^jV_{m,\mathbf a}}{\mathcal F_F^{j+1}V_{m,\mathbf a}}\\ &=\operatorname{im}\left( V_{m,\mathbf a}\cap H^0(Y,D_{m,\mathbf a}-jF) \longrightarrow H^0(F,(D_{m,\mathbf a}-jF)|_F) \right). \end{aligned}$ Here a section is first divided by its prescribed exceptional factor and then restricted. The displayed image is the definition. Replacing it by the complete space on the right requires a proof of surjectivity, or an asymptotic comparison such as Lemma 2.5. In particular, vanishing conditions at infinitely near points cannot be dropped merely because the ambient divisor class is known. Telescoping dimensions gives $N_m(W)=N_m(V)$ .

For a klt pair $(T,\Delta)$ , write $\Delta_Y$ for the strict transform of $\Delta$ , omitting $F$ if $F$ is a component on $T$ . We call $F$ of plt type if the above model exists and $(Y,\Delta_Y+F)$ is purely log terminal near $F$ . A divisor already on $T$ is included by taking the identity model and imposing the same plt condition. In particular, $F$ is normal. Adjunction defines its different by $(K_Y+\Delta_Y+F)|_F=K_F+\Delta_F.$ The pair $(F,\Delta_F)$ is klt. On a smooth model transverse to $F$ this is ordinary restriction of the other boundary components. On a quotient chart, the generic ramification indices contribute to $\Delta_F$ ; they must be retained even when the coarse variety $F$ is smooth.

If $F$ is not Cartier, the restriction in the refinement uses the rational divisor class furnished by [4, Lemma 2.7], whose sheaf is the quotient of the two adjacent divisorial sheaves. Thus the geometric quotient, its fractional fixed divisor, and its multiplication agree. Cartier powers can be used to compute section divisors, but they do not erase the other graded summands. For rational grading divisors one may first use a Cartier Veronese and then the comparison above. Under these conventions the refinements have bounded support and contain ample series whenever the original series does; this is [4, Example 2.15]. Repeated Cartier restrictions are covered by [4, Example 2.16].

For an extraction of a point $P$ of a smooth threefold $X$ , with $L=-K_X$ , the notation simplifies to $W_{m,j}=\mathcal F_G^jH^0(X,mL)/\mathcal F_G^{j+1}H^0(X,mL).$ Since $\sigma^*L|_G$ is trivial, the section class on $G$ is $-jG|_G$ . For a curve $B\subset G$ , its further refinement is $W^B_{m,j,q}=\mathop{\mathrm{gr}}_B^q W_{m,j}$ . Again this means restriction of the actual filtered subspace after removing $qB$ . The dimensions of all pieces still sum to $h^0(X,mL)$ .

2.4 Adjunction inequalities and strictness

Theorem 2.6 (The centre form of Abban–Zhuang adjunction). Let $(T,\Delta)$ be klt, let $V$ have bounded support and contain an ample series, and let $F$ be a divisor of plt type with model $\pi:Y\to T$ . Let $W$ be its refinement and $\Delta_F$ its different. Fix an irreducible subvariety $Z\subset T$ meeting $c_T(F)$ and an irreducible component $Z_0$ of their intersection. Put $b=\inf_{Z'\subset F,\ \pi(Z')=Z_0} \delta_{Z'}(F,\Delta_F;W).$ If $Z\subset c_T(F)$ , then $\delta_Z(T,\Delta;V)\ge \min\left\{\frac{A_{T,\Delta}(F)}{S(V;F)},b\right\}.$ If $Z\not\subset c_T(F)$ , then $\delta_Z(T,\Delta;V)\ge b$ . If $\delta_Z(T,\Delta;V)=b<A_{T,\Delta}(F)/S(V;F)$ and a valuation $v$ computes this threshold, then $c_Y(v)\not\subset F$ . For any component $U$ of $c_Y(v)\cap F$ whose image contains $Z_0$ , there is a valuation on $F$ centred at $U$ computing $\delta_{Z'}(F,\Delta_F;W)=b$ for every $Z'\subset U$ with $\pi(Z')=Z_0$ .

Proof. This is [4, Theorem 3.3]. We give the basis-divisor argument, including the term useful for strict inequalities. Choose a basis compatible with $F$ and with a chosen valuation $v$ satisfying $Z\subset c_T(v)$ . Its divisor satisfies $\pi^*D_m=S_m(V;F)F+\Gamma_m$ , where $F$ is not a component of $\Gamma_m$ and $\Gamma_m|_F$ is an $m$ -basis divisor of $W$ . If $\lambda_m$ is at most all the required local thresholds of this restricted divisor, inversion of adjunction gives log canonicity of $(Y,\Delta_Y+F+\lambda_m\Gamma_m)$ at the relevant centres meeting $F$ . Comparing its discrepancies with those of $(T,\Delta)$ yields $\label{az:eq:finite-retained} A_{T,\Delta}(v)\ge \lambda_m S_m(V;v) +\bigl(A_{T,\Delta}(F)-\lambda_m S_m(V;F)\bigr)v(F).$ When $Z\subset c_T(F)$ , also choose $\lambda_m\le A_{T,\Delta}(F)/S_m(V;F)$ . When $Z\not\subset c_T(F)$ , the centre of $v$ on $Y$ is not contained in $F$ , so $v(F)=0$ and this extra restriction is unnecessary. Proposition 2.4, with its uniform estimate, lets $\lambda_m$ tend to the stated minimum or to $b$ , respectively. This proves the inequalities and the limiting form of (2.1). Under the equality hypothesis the coefficient multiplying $v(F)$ is positive, so $v(F)=0$ . The existence of the computing valuation on $F$ is the equality clause of [4, Theorem 3.3], proved there by restricting the valuation ideals of $v$ and applying inversion of adjunction to the resulting graded sequence. ◻

In particular, if the relevant refined thresholds are at least $\lambda$ and $A_{T,\Delta}(F)\ge\lambda S(V;F)$ , then $\label{az:eq:retained} A_{T,\Delta}(v)-\lambda S(V;v) \ge\bigl(A_{T,\Delta}(F)-\lambda S(V;F)\bigr)v(F).$ This is useful even when the refined threshold equals one. If $\lambda=1$ , the coefficient on the right is positive, and $c_Y(v)\subset F$ , then $v(F)>0$ and $A_{T,\Delta}(v)>S(V;v)$ . This is a strict inequality for that valuation; it does not by itself give a uniform positive lower bound as $v$ varies. For a point extraction, valuations centred at the extracted point have $v(F)>0$ . Valuations centred on a curve or divisor through that point have $v(F)=0$ and require their own estimate. This distinction is used in the final assembly of the Family 1.9 proof.

Proposition 2.7 (Abban–Zhuang refinement). Let $X$ be a smooth Fano threefold, and suppose that $\sigma:\widehat X\to X$ extracts a plt-type divisor $G$ over a point $P$ . Let $W$ be the refinement of the anticanonical series. Then $\label{f25:eq:first-refinement} \delta_P(X)\ge\min\left\{\frac{A_X(G)}{S_X(G)},\delta(G,\Delta_G;W)\right\}.$ Suppose that $B\subset G$ is rational Cartier, $B\not\subset\mathop{\mathrm{Supp}}\Delta_G$ , and $(G,\Delta_G+B)$ is plt along $B$ . Put $(K_G+\Delta_G+B)|_B=K_B+\Delta_B$ . For every point $Q\in B$ , $\label{f25:eq:second-refinement} \delta_Q(G,\Delta_G;W)\ge \min\left\{\frac1{S(W;B)}, \frac{1-\operatorname{coeff}_Q\Delta_B}{S(W^B;Q)}\right\}.$ For a plt-type extraction of a divisor $K$ over a point $R$ of a klt surface pair $(T,\Delta)$ , with any series satisfying the preceding hypotheses, the corresponding formula is $\label{f25:eq:surface-extraction} \delta_R(T,\Delta;W)\ge \min\left\{\frac{A_{T,\Delta}(K)}{S(W;K)}, \inf_{Q\in K}\frac{1-\operatorname{coeff}_Q\Delta_K}{S(W^K;Q)}\right\}.$ In all three formulas the series are the actual refinements with the original total-degree normalization, and they have bounded support and contain ample series.

Proof. Apply Theorem 2.6. For the first formula every centre on $G$ maps to $P$ , so the second term is the global threshold on $G$ . For the second formula $A_{G,\Delta_G}(B)=1$ , and the normal curve $B$ is smooth. Its only divisorial valuation centred at $Q$ , up to scaling, is $\mathop{\mathrm{ord}}_Q$ ; its discrepancy is $1-\operatorname{coeff}_Q\Delta_B$ . For the third formula every point of the exceptional curve must be considered. This proves the three assertions. The rational-divisor formulation is also given in [17, Theorem 3.20]. ◻

For a smooth surface $S\subset X$ and a smooth curve $C\subset S$ through $p$ , repeated restriction gives the familiar bound by the minimum of the three ratios attached to $S$ , $C$ and $p$ . The first denominator is $S_X(S)$ , the second is the expected order of the actual restricted series along $C$ , and the third is the expected point order of its further restriction. When a quotient singularity occurs, replace the numerator one at $p$ by its discrepancy for the different. A longer Cartier flag is handled in precisely the same way by its lexicographic filtration and repeated inversion of adjunction; see [4, Theorem 3.5]. The surface intersection formulas in the next section compute these denominators.

If the curve used for refinement already has coefficient $b<1$ in the surface boundary, write $\Delta=bB+\Delta'$ with $B\not\subset\mathop{\mathrm{Supp}}\Delta'$ . The first numerator is then $1-b$ , and the different is defined from $(G,B+\Delta')$ , not from $(G,(1+b)B+\Delta')$ . This is the divisor-on-the-variety case of Theorem 2.6.

Lemma 2.8 (A point that may depend on the basis). Let $C$ be an irreducible curve on a smooth variety $X$ , let $S\subset X$ be a smooth divisor not containing $C$ , and let $\Sigma\subset C\cap S$ be nonempty. Let $V$ be a complete ample series. Suppose that a fixed flag filtration refines the order along $S$ , and that, for every basis divisor compatible with this filtration, the decomposition $D_m=s_mS+D'_m$ has the following property: for some $p_m\in\Sigma$ , the pair $(S,\lambda_mD'_m|_S)$ is log canonical at $p_m$ , where $\lambda_m\ge0$ , $\lambda_m s_m\le1$ and $\lambda_m\to\lambda$ . Then $\delta_C(X;V)\ge\lambda$ .

Proof. Fix $v$ whose centre contains $C$ , and choose a basis compatible with $v$ and the single flag filtration. By inversion of adjunction, $(X,S+\lambda_mD'_m)$ is log canonical near its associated point $p_m$ . Lowering the coefficient of $S$ to $\lambda_m s_m$ proves the same for $(X,\lambda_mD_m)$ . If $A_X(v)<\lambda_m v(D_m)$ , its closed non-log-canonical locus contains $c_X(v)$ and therefore every point of $C$ , a contradiction. Hence $A_X(v)\ge\lambda_mS_m(V;v)$ . Take the limit. No point of $\Sigma$ has to work for every basis divisor. This is the finite-level form of the flag argument used in Lemma ??. ◻

Example 2.9 (An ordinary point blow-up). Let $T=\mathbb P^2$ , $M=\mathcal O_{\mathbb P^2}(d)$ , with $d$ a positive integer, and blow up $p$ with exceptional curve $E$ . In affine coordinates centred at $p$ , homogeneous polynomials of degree $md$ have arbitrary initial homogeneous forms of degree $j$ for every $0\le j\le md$ . Hence the actual refined space is $W_{m,j}=H^0(E,\mathcal O_E(j))$ , of dimension $j+1$ . Summing over $j$ recovers $N_m=\binom{md+2}{2}$ . It follows that $S_M(E)=\lim_m\frac{\sum_{j=0}^{md}j(j+1)}{mN_m}=\frac{2d}{3}, \qquad S(W;Q)=\lim_m\frac{\sum_{j=0}^{md}j(j+1)/2}{mN_m}=\frac d3$ for every $Q\in E$ . Since $A_T(E)=2$ and the different is zero, Proposition 2.7 gives $\delta_p(T;M)\ge3/d$ . A line through $p$ has discrepancy one and expected order $d/3$ , so equality holds. This example exhibits the two different discrepancies, two and one, in a point blow-up, and explains why an average over one fixed $j$ would have the wrong normalization.

2.5 Comparison with a complete series

There is a useful special case in which an entire multigraded calculation reduces to one complete linear series. For each summand write its linear system as a fixed divisor $F_{m,\mathbf a}$ plus a system with no fixed divisorial component. Its average fixed divisor is the sum of $F_{m,\mathbf a}$ , weighted by $\dim V_{m,\mathbf a}/(mN_m(V))$ . Suppose its limit $F(V)$ is supported on finitely many prime divisors. Fix a big line bundle $H$ . We say that the moving series are asymptotically complete multiples of $H$ if, for every rational interior index $\boldsymbol\gamma$ on the support slice, their moving spaces lie in complete systems of divisors numerically equivalent to $\ell_{m,\boldsymbol\gamma}H$ , with $\ell_{m,\boldsymbol\gamma}$ positive integral, and their dimension divided by $h^0(T,\ell_{m,\boldsymbol\gamma}H)$ tends to one along sufficiently divisible $m$ . These are the conditions called almost completeness in [4, Definition 2.27]; the actual moving spaces, not just their numerical classes, enter the condition. The limiting divisors may have real coefficients. Log discrepancies, log canonicity and the valuative definition of the threshold are used for these real boundaries by the same formulas.

Proposition 2.10. Under these conditions, and assuming bounded support and the ample-series condition, the average moving class is $\mu H$ for a nonnegative real number $\mu$ , the divisor $F(V)$ is real Cartier, and $S(V;v)=\mu S_H(v)+v(F(V))$ for every divisorial valuation $v$ . Consequently, whenever $\Delta+\lambda F(V)$ is an effective klt boundary and $\mu>0$ , an estimate $\delta_Z(T,\Delta+\lambda F(V);\mu H)\ge\lambda$ implies $\delta_Z(T,\Delta;V)\ge\lambda$ .

Proof. The expected-order identity, including the existence of the average moving class, is [4, Lemma 2.29]. Fixed divisors contribute their actual valuation; asymptotic completeness supplies the expected order of the complete moving series. One way to see why the dimension condition is relevant is Lemma 2.5: an asymptotically negligible loss of sections changes the normalized first moment by an asymptotically negligible amount when the orders have a linear bound. Integrating over the bounded support gives the stated coefficient $\mu$ . For the last assertion, the assumed inequality says $A_{T,\Delta}(v)-\lambda v(F(V))\ge\lambda\mu S_H(v)$ for each tested valuation. Substitute the expected-order identity and take the infimum. ◻

Example 2.11 (Restriction to an anticanonical surface). Let $X$ be a smooth Fano threefold with base point free $H=-K_X$ , and let $S\in|H|$ be smooth. Its refined summands are the images of $H^0(X,(m-j)H)$ on $S$ , for $0\le j\le m$ . They are complete for $m-j\gg0$ by Serre vanishing. Their asymptotic dimensions are $H^3(m-j)^2/2$ , whereas $h^0(X,mH)\sim H^3m^3/6$ . Thus their average moving class is $3\int_0^1(1-u)^3\,du\ H|_S=\frac34 H|_S,$ and $S(W;v)=\frac34 S_{H|_S}(v)$ . This is the coefficient used in the final remark of Part ??. The calculation does not assert that the resulting surface estimate is strong enough for K-stability; the choice of an auxiliary surface affects the strength of the bound.

2.6 An isolated point and the multiplier ideal

The following supplement is used when estimates have already excluded all positive-dimensional centres through a point. It is distinct from restriction to a surface.

Lemma 2.12. Let $T$ be smooth and projective, let $D$ be effective rational Cartier, and suppose that $-(K_T+D)$ is ample. If $(T,D)$ is klt in a punctured neighbourhood of $p$ , then every divisorial valuation $v$ centred at $p$ satisfies $A_T(v)\ge\frac{\nu(v)}{\nu(v)+1}v(D), \qquad \nu(v)=\frac{A_T(v)}{v(\mathfrak m_p)}.$

Proof. This is the multiplier-ideal argument of [4, Lemma 4.24]. If $(T,D)$ is klt at $p$ , the assertion follows from $A_T(v)>v(D)$ . Otherwise put $\mathcal J=\mathcal J(T,D)$ . Nadel vanishing gives $H^1(T,\mathcal J)=0$ and hence a surjection $H^0(T,\mathcal O_T)\to H^0(T,\mathcal O_T/\mathcal J)$ . The point $p$ is an isolated component of its support, so its local quotient is a direct summand of these global sections. It has length at most one, and is nonzero. Therefore $\mathcal J_p=\mathfrak m_p$ . The valuation inequality for a multiplier ideal gives $v(\mathfrak m_p)\ge v(D)-A_T(v)$ . Substituting $v(\mathfrak m_p)=A_T(v)/\nu(v)$ gives the result. ◻

Corollary 2.13. Let $X$ be a smooth Fano threefold with base point free $H=-K_X$ . Suppose that every valuation whose positive-dimensional centre contains $p$ has $A_X(w)\ge\chi S_X(w)$ , where $1<\chi<4/3$ . Then, for every divisorial valuation $v$ centred at $p$ , $\frac{A_X(v)}{S_X(v)}\ge \chi\frac{\nu(v)}{\nu(v)+1}, \qquad \nu(v)=\frac{A_X(v)}{v(\mathfrak m_p)}.$

Proof. Choose a smooth $T\in|H|$ avoiding $p$ and a basis compatible with $T$ and $v$ . Write $D_m=a_mT+\Gamma_m$ , where $a_m=S_m(H;T)\to1/4$ and $\Gamma_m\sim_{\mathbb Q}(1-a_m)H$ is effective. Fix a rational number $0<\lambda<\chi$ . The uniform estimate in Proposition 2.4 implies, for large $m$ , that $(X,\lambda\Gamma_m)$ is klt at the generic point of every positive-dimensional centre containing $p$ . Its non-klt locus is closed, so it is klt in a punctured neighbourhood of $p$ . Moreover $-(K_X+\lambda\Gamma_m)$ is ample because $\lambda(1-a_m)<1$ . Apply Lemma 2.12. Since $T$ avoids $p$ , one has $v(\Gamma_m)=v(D_m)=S_m(H;v)$ . Let $m$ tend to infinity and then let $\lambda$ increase to $\chi$ . This is the argument used in Lemma ??. ◻

2.7 Fractional section divisors

Equalities of rational divisor classes do not identify the actual boundary or the zero divisor of a specified section. The following convention is used on every weighted exceptional surface in the manuscript.

Lemma 2.14 (Divisors of weighted forms). Let $T=\mathop{\mathrm{Proj}}\mathbb C[x,y,z]$ for positive primitive weights, and let $h$ be a nonzero homogeneous form of weight $j$ . Choose $r$ for which $\mathcal O_T(rj)$ is invertible, and define $D(h)=r^{-1}\operatorname{div}_0(h^r)$ . This effective rational divisor is independent of $r$ and additive under multiplication. Forms $h_1,h_2$ of the same weight satisfy $D(h_1)-D(h_2)=\operatorname{div}(h_1/h_2)$ . If a prime curve $B$ is not a component of $D(h)$ , restriction to $B$ is computed by restricting a Cartier power and dividing by that power.

Proof. Pass to a common multiple of two possible choices of $r$ . Both give the same divisor after multiplication by that common multiple; this proves independence and additivity. The ratio of two forms of equal weight is a rational function on $T$ , and the same Cartier-power computation proves the difference formula. The final assertion is the definition of restriction of an effective rational Cartier divisor not containing $B$ . Removing the exceptional factor in a weighted blow-up gives exactly these divisors. ◻

On a fixed weight space, the valuation of a section is therefore an ordinary valuation of its rational representative plus a fixed shift. Compatible bases remain legitimate. In Family 2.5, type $D_5$ , the relation $z^2=W$ gives $D(z)=\frac12(W=0)$ on the coarse surface. In type $E_6$ , the relations $y^3=W$ and $z^2=V$ give $D(y)=\frac13(W=0)$ and $D(z)=\frac12(V=0)$ . Here $y,z$ are weighted sections; they are not rational functions on the coarse surface. These fractional factors enter the point orders of the refined curve series.

For clarity, representatives of the positive normal class $-G|_G$ in these two cases are $D_1=\frac12(W=0)-(V=0)\sim_{\mathbb Q}H_G/2$ and $D_1=\frac13(W=0)-\frac12(V=0)\sim_{\mathbb Q}H_G/6$ , respectively. The semi-invariants $\theta=z/y$ and $\theta=y/z$ have native weight one. Thus a form $h$ of native weight $j$ defines a rational function $h/\theta^j$ on the coarse surface and satisfies $D(h)=jD_1+\operatorname{div}(h/\theta^j)$ . This realizes all native weights in the same rational-divisor convention. The divisor $D_1$ specifies a section space and need not be effective. The zero divisor $D(h)$ is effective, whereas the different is a separate boundary determined by adjunction.