18.39 Singularities along a section over a divisorDFED5BDE8678

Proof. For $d=0$, the manifold $M$ is a finite set and the dimension is at most its number of points. Let $d\geq 1$. Fix smooth Hermitian metrics on $E$ and $F$, a smooth positive volume form $\mathrm{d}V$, and finitely many coordinate charts with holomorphic frames, together with smaller subcharts that still cover $M$. For $|j|\leq C_0m$, the local weight of the metric on $E^ m\otimes F^ j$ has second derivatives bounded by $Cm$. At a point $p$ of a smaller subchart, multiply the frame by the exponential of a holomorphic affine function to remove the constant and real-linear terms of this weight at $p$. The new weight vanishes at $p$ and is bounded by $Cm|z-p|^2$ on a fixed neighborhood. A ball of radius $\varepsilon m^{-1/2}$ about $p$ lies in the chart, with $\varepsilon >0$ independent of $p$, $m$ and $j$, and on this ball the weight is bounded. The submean inequality for the holomorphic coefficient of a section $s$ on this ball gives

where all constants depend only on the fixed charts, metrics, volume and $C_0$. Evaluation at $p$ has a one-dimensional target, so for an $L^2$-orthonormal basis $s_1,\ldots ,s_ N$ of $H^0(M,E^ m\otimes F^ j)$ its squared norm is $\sum _{\ell =1}^ N|s_\ell (p)|^2\leq Cm^ d$. Integrating over $M$ gives $N\leq C\operatorname {Vol}(M)m^ d$. $\square$

Proof. By the Poincaré–Lelong formula, $\theta +\mathrm{dd}^{\mathrm{c}}\psi =\alpha +c[Z]\geq 0$. The divisorial measure identity (11.41.2), applied with the residual form $\alpha >0$ and the bounded part zero, gives $\mathrm{MA}_\theta (\psi )=\alpha ^ q$, which proves the mass formula.

Every section of $\pi ^*A^ m$ descends by proposition 18.36.1. In the $r+1$ normal variables $(x,w)$, lemma 5.37.1 gives

The local coefficient $g(x,z')$ of a pulled-back section does not depend on $w$. Expanding $g$ in powers of $x$, we see that $g$ lies in this ideal exactly when the coefficients of $x^ j$ vanish for $j<k_ m$, that is, when $x^{k_ m}$ divides $g$. This is an identity of germs in all tangential variables, so it holds at every point of $Z$. Outside $Y$ the function $\varphi $ is locally bounded. On the base, the same lemma in the single normal variable $x$ gives $\mathcal I(m\psi )=\mathcal O_ B(-\ell _ mZ)$. These facts prove the two identities and the inclusion.

For $k_ m\leq j<\ell _ m$, the restriction sequence along $Z$,

shows that passing from vanishing order $j$ to order $j+1$ along $Z$ loses at most $h^0(Z,(A^ m(-jZ))|_ Z)$ dimensions. Apply lemma 18.39.1 on $Z$ with $E=A|_ Z$ and $F=\mathcal O_ B(-Z)|_ Z$; since $0\leq j\leq cm$, each loss is at most $Cm^{q-1}$, uniformly in $j$. There are at most $r$ steps, which proves the estimate. When $q=1$, the divisor $Z$ is a finite set and each restriction space has dimension $\# Z$, which gives the last assertion directly. $\square$

Proof. Use homogeneous coordinates $[Z_0:\cdots :Z_ q]$ on the base and $[W_0:\cdots :W_ r]$ on the fiber. Let $Z=\{ Z_0=0\} $ and $Y=Z\times \{ [1:0:\cdots :0]\} $, and let $\omega _ B,\omega _ F$ be the Fubini–Study forms normalized to have top integrals one. Define

The function $\varphi $ is smooth off $Y$ and satisfies (18.39.2). Over a common denominator, the numerator of $R_ B+R_ F$ is

whose logarithm is plurisubharmonic in each affine chart. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -c(\omega _ B+\omega _ F)$. Choose an integer $b\geq c$ and set $\omega =k\omega _ B+b\omega _ F$. This is a Hermitian form, and $\omega +\mathrm{dd}^{\mathrm{c}}\varphi \geq (k-c)\omega _ B+(b-c)\omega _ F\geq 0$. On the base, $\theta =k\omega _ B$, $\Theta _ Z=\omega _ B$ and $\alpha =(k-c)\omega _ B>0$.

Put $\ell _ m=\lfloor mc\rfloor $. Divisibility of homogeneous polynomials by powers of $Z_0$ gives the exact dimensions

For large $m$, the two residual degrees differ by $r$ and equal $(k-c)m+O(1)$. Subtracting the two binomial coefficients, which are polynomials of degree $q$ in the residual degrees, gives the asserted leading coefficient and remainder. For $q=1$ the difference is $\ell _ m-\max \{ \ell _ m-r,0\} =\min \{ r,\ell _ m\} $ for every $m$. $\square$

Proof. As in proposition 18.36.1, fiber integration identifies the norm on $X$ with the norm on $V_ m\subseteq H^0(B,A^ m)$, so the kernel measure on $X$ is the lift of the kernel measure of $V_ m$ on $B$. By lemma 18.37.1, theorem 18.39.3, the latter differs from the kernel measure of $W_ m$ by a positive measure of total mass $q!(\dim V_ m-\dim W_ m)/m^ q=O(m^{-1})$, and so do the lifts. Since $B$ is Kähler and $\psi $ has positive non-pluripolar mass, the equilibrium theorem used in the proof of theorem 18.36.3 applies to $W_ m$ and gives the weak limit. By lemma 18.37.1, the determinant functionals of $V_ m$ and $W_ m$ differ by at most $O(m^{-1})\| v\| _\infty $, and corollary 18.35.6, applied on $B$ with the identity modification, gives the limit of the functionals; its dominated convergence bound is $\| v\| _\infty $ times the bounded sequence of $q$-normalized dimensions. Finally, the $q$-normalized masses are bounded, so multiplying them by $(q+r)!m^{-r}/q!$ proves the last assertion. $\square$

Proof. After integrating over the angles, substitute $x_0=|z|^{2p}$ and $x_ j=|w_ j|^{2b_ j}$. Up to a positive constant, the integral becomes

over a box near the origin. This box contains a small simplex $\{ \sum _ jx_ j<\delta \} $ and is contained in a larger one. On either simplex use the coordinates $\rho =\sum _ jx_ j$ and $x_ j=\rho y_ j$. The integral over $y$ is finite and positive, because all its exponents exceed $-1$, and the integral over $\rho $ is

This gives the threshold and the logarithmic divergence at equality. Bounded factors in the weight, holomorphic units and additional tangential variables do not affect this local test. $\square$

Proof. By the Poincaré–Lelong formula, $\theta +\mathrm{dd}^{\mathrm{c}}\psi =\beta +c[D]$. Off $D$, locality identifies the non-pluripolar top power with $\beta ^ q$; the non-pluripolar product puts no mass on $D$, and neither does the smooth measure $\beta ^ q$. Hence $\mathrm{MA}_\theta (\psi )=\beta ^ q$, whose mass is finite and positive because $\beta >0$ and $B$ is compact. The one-variable integrability condition $\ell +1>mc$ gives $\mathcal I(m\psi )=\mathcal O_ B(-L_ mD)$, also when $mc$ is an integer. Since $u$ is bounded, the global lower bound implies $\varphi \geq \psi \circ \pi -C''$. Hence integrability on the base gives integrability of the pullback in every bounded submersion chart, which proves $W_ m\subseteq V_ m$.

For the upper inclusion, take a nonzero section in $V_ m$ and let $\ell $ be its vanishing order along $D$; note that $D$ is irreducible, being smooth and connected. In the polydisk of the upper bound, the coefficient of the section is $z^\ell g(z,t)$, where $g(0,t)$ does not vanish identically on the open subset of $D$, since otherwise the identity theorem would increase the order along $D$. Shrink the polydisk about a point with $w=0$ at which $g(0,t)\ne 0$, so that $g$ becomes a holomorphic unit. The upper bound on $\varphi $ and lemma 18.39.7 then force $\ell +1>mc-a$. For an integer $\ell \geq 0$ this is equivalent to $\ell \geq K_ m$, which proves the upper inclusion.

Since $\lfloor mc\rfloor -\lfloor mc-a\rfloor \leq \lceil a\rceil $, we have $0\leq L_ m-K_ m\leq \lceil a\rceil $. Use the restriction sequences along $D$ from the proof of theorem 18.39.3 for $K_ m\leq j<L_ m$. Each step loses at most $h^0(D,(A^ m(-jD))|_ D)$ dimensions, and by lemma 18.39.1 this is at most $Cm^{q-1}$, uniformly for $0\leq j\leq cm$. Since the number of steps is bounded, the estimate follows. For $q=1$ the connected divisor $D$ is a single point, and each restriction space has dimension one. $\square$

Proof. The local model is bounded below by $c\log |z|^2$ up to a constant. Near points of $\pi ^{-1}(D)$ outside $Z$, the function $\varphi $ is locally bounded, so the same lower bound holds there; away from $\pi ^{-1}(D)$ both functions are locally bounded. By compactness, finitely many charts give the global lower bound. The upper bound holds in any chart of the model. Conversely, if a section is divisible by $z^{K_ m}$, its residual coefficient is bounded and holomorphic in each chart of the model, and the strict inequality $K_ m+1>mc-a$ together with lemma 18.39.7 gives integrability there. Outside $Z$ the function $\varphi $ is locally bounded, which proves the reverse inclusion. $\square$

Proof. Locally, the complex Hessian of the logarithm of a sum of two squared norms of holomorphic sections is bounded below by a convex combination of the negatives of their curvature forms. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -\tfrac 12\omega _ B-\tfrac 34\omega _ F$ off the pole set, and across it by the extension theorem after a quadratic correction. Thus $\varphi \in \mathrm{PSH}(X,\omega )$. The pole set is $D\times \{ o\} $, along which $\varphi $ has the exact local model of corollary 18.39.9. Here $u=0$, $\beta =\tfrac 12\omega _ B$, and $K_ m=L_ m-1$ for $m\geq 2$, while $K_1=L_1=0$. The spaces $V_ m$ and $W_ m$ therefore correspond to homogeneous polynomials of degrees $m-L_ m+1$ and $m-L_ m$, and since $m-L_ m=\lceil m/2\rceil $, the difference of their dimensions is the displayed binomial coefficient. $\square$

Proof. For $d=0$ the dimension is at most the number of points of $M$. For $d\geq 1$, use fixed smooth metrics, a smooth positive volume form and finitely many framed charts as in the proof of lemma 18.39.1. The local weight of the metric is $m\phi _ E+\sum _ i j_ i\phi _{F_ i}$, whose second derivatives are bounded by $Cm$ uniformly over the stated range. After its constant and real-linear terms at a point have been removed by a holomorphic affine change of frame, it is bounded by a fixed constant on a ball of radius $\varepsilon m^{-1/2}$ about that point. The submean inequality then gives $|s(x)|^2\leq Cm^ d\| s\| _{L^2}^2$, uniformly in $x$, $m$ and the integers $j_ i$. Since evaluation has a one-dimensional target, an orthonormal basis satisfies $\sum _\ell |s_\ell (x)|^2\leq Cm^ d$, and integration gives the estimate. The constants depend only on the fixed bundles, metrics, charts, volume and $C_0$. $\square$

Proof. By the Poincaré–Lelong formula, $\theta +\mathrm{dd}^{\mathrm{c}}\psi =\beta +\sum _ i c_ i[D_ i]$. Off the divisors the non-pluripolar measure equals $\beta ^ q$, and neither measure charges their union. Thus $\mathrm{MA}_\theta (\psi )=\beta ^ q$, with finite positive total mass.

Near a point of an intersection, the divisors through that point are distinct coordinate hyperplanes $\{ z_ i=0\} $. If a holomorphic germ is divisible by each corresponding $z_ i^{L_{m,i}}$, then it is divisible by their product; its residual coefficient is bounded, and since $L_{m,i}+1>mc_ i$, Fubini's theorem gives integrability. Conversely, integrability at generic nearby points of each coordinate hyperplane forces divisibility by $z_ i^{L_{m,i}}$, by the one-variable threshold, and the divisor of the germ then gives divisibility throughout the chart. This proves

Since $u$ is bounded, the global lower bound implies $\varphi \geq \psi \circ \pi -C'$, and integration in submersion coordinates gives $W_ m\subseteq V_ m$. For a nonzero section in $V_ m$, the polydisk over $D_ i$ on which the upper bound holds detects the generic vanishing order of the section along the irreducible divisor $D_ i$. The argument with a unit coefficient in the proof of theorem 18.39.8, together with the strict integrability threshold of lemma 18.39.7, shows that this order is at least $K_{m,i}$. Imposing all these orders proves the upper inclusion.

To compare dimensions, raise the orders from $(K_{m,i})_ i$ to $(L_{m,i})_ i$, one divisor and one unit at a time. At each step, the restriction sequence loses at most

dimensions. By lemma 18.39.12 with $E=A|_{D_ i}$ and $F_ j=\mathcal O_ B(-D_ j)|_{D_ i}$, this is $O(m^{q-1})$ uniformly over all steps, and there are at most $\sum _ i\lceil a_ i\rceil $ steps. When $q=1$, the distinct connected divisors are points, and each restriction space has dimension one. $\square$

Proof. By the calculation in example 18.39.10, the complex Hessian of each summand of $\varphi $ is bounded below by $-\tfrac 12\omega _{B,i}-\tfrac 34\omega _{F,i}$, also across its pole set by the extension theorem after a quadratic correction. Summing proves $\varphi \in \mathrm{PSH}(X,\omega )$. The global lower bound follows from $R_ i^2+T_ i^3\geq R_ i^2$. For the upper bound over $D_ i$, away from the other divisors, place the $i$th fiber coordinate at its marked point; all the other summands are smooth there. The residual is $\beta =\tfrac 12\sum _ i\omega _{B,i}>0$, whose top integral is $q!/2^ q$.

The theorem gives the required generic orders along every $D_ i$. Conversely, near a point where several of the pairs $(R_ i,T_ i)$ vanish, the density $\mathrm{e}^{-m\varphi }$ is comparable to the product of the factors $(|z_ i|^4+|w_ i|^6)^{-m/4}$ in disjoint pairs of variables. A descended coefficient divisible by every $z_ i^{K_ m}$ has a bounded holomorphic residual. Each factor is integrable because $K_ m+1>m/2-2/3$, and Fubini's theorem gives integrability at the intersections; elsewhere the same argument uses fewer factors. This proves the identity for $V_ m$, and the theorem gives $W_ m$. A line bundle $\mathcal O_ B(d,\ldots ,d)$ with $d\geq 0$ has $(d+1)^ q$ independent sections, the products of monomials in the separate factors. Since $K_ m=L_ m-1$ for $m\geq 2$ and $m-L_ m=\lceil m/2\rceil $, subtraction gives the displayed defect and its leading term. $\square$

Proof. By the Poincaré–Lelong formula, $\theta '+\mathrm{dd}^{\mathrm{c}}\psi '=\beta +\sum _ i c_ i[D_ i]$. Thus $\psi '\in \mathrm{PSH}(Y,\rho ^*\theta )$, and proposition 5.43.18 gives the unique potential $\psi $ on $B$ with $\psi \circ \rho =\psi '$. On $Y$, locality off the union of the divisors and the fact that the non-pluripolar measure does not charge this union give $\mathrm{MA}_{\theta '}(\psi ')=\beta ^ q$. The measure identity in proposition 18.35.1 then gives the pushforward formula and the mass.

Lemma 5.43.22, applied first with weight zero and then with weight $m\psi $, gives the identification of the full section spaces and the formula for $T_ m(W_ m)$; here $\mathcal I(m\psi ')=\mathcal O_ Y(-\sum _ iL_{m,i}D_ i)$ by the integrability computation for simple normal crossings in the proof of theorem 18.39.13. Since $u$ is bounded, the global lower bound gives $\varphi '\geq \psi '\circ \pi '-C'$. As $\rho \circ \pi '=\pi \circ \mu $ and $\mu $ is surjective, this descends to $\varphi \geq \psi \circ \pi -C'$, and integrability in bounded submersion charts proves $W_ m\subseteq V_ m$.

For $s\in V_ m$, the transformation of multiplier ideals under $\mu $ gives

Hence, in each polydisk where the upper bound holds, the descended coefficient of $T_ m(s)$ vanishes to order at least $K_{m,i}$ along $D_ i$, by lemma 18.39.7 and the argument with a unit coefficient in the proof of theorem 18.39.8. This is the vanishing order of $T_ m(s)$, including the Jacobian factor, and it proves the upper inclusion.

The chain of restrictions along the divisors on $Y$ has at most $\sum _ i\lceil a_ i\rceil $ steps. On $D_ i$, each target is $(\rho ^*A|_{D_ i})^ m$ twisted by powers of the bundles $\mathcal O_ Y(-D_ j)|_{D_ i}$ in a linear range and by the fixed bundle $R_\rho |_{D_ i}$. Including the latter in lemma 18.39.12 with exponent one, we see that every target has dimension $O(m^{q-1})$, uniformly. Summing gives the estimate. For $q=1$ each divisor is a point and every target has dimension one. $\square$

Proof. Over a common denominator, $R^2+T^3$ is a sum of squared moduli of holomorphic coefficients of sections of $\mathcal O_ X(2,3)$, divided by the corresponding metric factor. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -\tfrac 12\omega _ B-\tfrac 34\omega _ F$, also across the common zero set by the extension theorem.

Let $\rho :Y\to B$ be the blowup of $p$, with exceptional divisor $E$, and let $\tau :Y\to \mathbb P^{q-1}$ be the extension to $Y$ of the projection $[Z_0:\cdots :Z_ q]\mapsto [Z_1:\cdots :Z_ q]$. Since $(\rho ,\tau )$ is an embedding, the forms

satisfy $\beta >0$. The divisor bundle is $\mathcal O_ Y(E)=\rho ^*\mathcal O_ B(1)\otimes \tau ^*\mathcal O_{\mathbb P^{q-1}}(-1)$; its quotient metric has curvature $\alpha -\gamma $ and satisfies $R\circ \rho =\| s_ E\| ^2$. Thus the theorem applies with the single divisor $E$, $u=0$, $c=1/2$, $a=2/3$ and residual $\beta $. Along $E\times \{ o\} $ the pulled-back selecting potential has the exact local model $\tfrac 14\log (|z|^4+|w|^6)+O(1)$, and it is locally bounded elsewhere. If $T_ m(s)$ is divisible by $z^{K_ m}$, where $K_ m=\max \{ \lfloor m/2-2/3\rfloor ,0\} $, then lemma 18.39.7 gives integrability in every chart of this model; this gives the inclusion opposite to the upper inclusion of the theorem.

In the blowup coordinates $Z_1/Z_0=z$ and $Z_ j/Z_0=zt_ j$ for $j\geq 2$, the Jacobian of $\rho $ is $z^{q-1}$. Hence, if a section $s$ of $A^ m$ vanishes to order $d$ at $p$, then $T_ m(s)$ vanishes to order $d+q-1$ generically along $E$. The two divisorial thresholds $\lfloor m/2\rfloor $ and $K_ m$ therefore give exactly the orders $\ell _ m$ and $k_ m$ above. Homogeneous polynomials of degree $m$ realize every jet of total degree at most $m$ at $p$. For $m\geq 2q$ we have $k_ m=\ell _ m-1$, and the additional jets of degree $\ell _ m-1$ are spanned by $\binom {\ell _ m+q-2}{q-1}$ monomials, which gives the defect. For $m<2q$ both orders vanish.

Finally, in cohomology $[\beta ]=\rho ^*c_1(A)-\tfrac 12[E]$. The intersection numbers containing both $\rho ^*c_1(A)$ and $[E]$ vanish, because $\rho ^*c_1(A)$ restricts to zero on $E$. The normal bundle of $E\simeq \mathbb P^{q-1}$ is $\mathcal O_ E(-1)$, so $\int _ Y[E]^ q=(-1)^{q-1}$, while $\int _ Y(\rho ^*c_1(A))^ q=1$. Expanding gives $\int _ Y\beta ^ q=1-2^{-q}$, which is the mass of $\psi $ by the theorem. $\square$

Proof. In each case the reference current is $\beta +\sum _ i c_ i[D_ i]$, which is positive. Locality off the union of the divisors, together with the fact that the non-pluripolar measure does not charge this union, gives the non-pluripolar measure $\beta ^ q$ on $M$. Its mass is finite because $\beta $ is smooth and $M$ is compact, and positive by assumption. In the case of a modified base, the potential descends and its measure is $\rho _*(\beta ^ q)$, by the same descent and measure identities as before.

The multiplier ideal on the base depends only on the logarithmic divisorial orders and the bounded smooth term. The lower inclusion uses the global lower bound for the selecting potential, and the upper inclusion uses the anisotropic threshold in a polydisk where the upper bound holds; neither uses strict positivity of $\beta $. The restriction estimates concern fixed line bundles, whatever their curvature, and the Jacobian and the fixed canonical twist in the case of a modified base are unchanged. Hence the inclusions and dimension bounds in all three proofs remain valid. The reverse inclusion for the exact anisotropic model uses only its local integrability test and the local boundedness off the pole set, so it also remains valid. $\square$

Proof. Since $(\rho ,\tau )$ is an embedding, $\kappa >0$. The quotient metric satisfies

as in example 18.39.17. The same calculation over a common denominator, now with coefficient $1/2$, gives

off the pole set $E\times \{ o\} $, and across it by the extension theorem. Hence $\omega +\mathrm{dd}^{\mathrm{c}}\varphi \geq \pi ^*\gamma +\tfrac 12\omega _{\mathbb P^1}\geq 0$. Along the smooth pole set $E\times \{ o\} $, the function $\varphi $ has the exact local model $\tfrac 12\log (|z|^4+|w|^6)+O(1)$, and it is locally bounded elsewhere. It also satisfies the global lower bound $\varphi \geq \log R$.

We may therefore take $c=1$, $u=0$, $p=2$, $b_1=3$ and $a=2/3$, with residual $\beta =\theta -\eta =k\alpha \geq 0$. Its top integral is $k^ q$, since the normalized Fubini–Study form has top integral one and $\rho $ is a modification. Corollary 18.39.19, Corollary 18.39.9 therefore give

Let $t\geq 0$ be an integer. A section of $\rho ^*\mathcal O(km)\otimes \mathcal O_ B(tE)$, divided by $s_ E^ t$, gives a holomorphic section of $\mathcal O(km)$ on $\mathbb P^ q\setminus \{ p\} $, which extends across $p$ by Hartogs' theorem since $q\geq 2$. Pulling the extension back and multiplying by $s_ E^ t$ recovers the original section off $E$, hence everywhere by the identity theorem. Thus every section of $A^ m$ has the form $s_ E^ m\rho ^*s$, and the natural embeddings of $W_ m$ and $V_ m$ into $H^0(B,A^ m)$ both have this image. Its dimension is the number $\binom {km+q}{q}$ of monomials of degree $km$.

Finally, let $C\subset E\simeq \mathbb P^{q-1}$ be a projective line; it is contracted by $\rho $. For every smooth real function $u$, Stokes' theorem gives $\int _ C(k\alpha +\mathrm{dd}^{\mathrm{c}}u)=0$, whereas a strictly positive form has positive integral over $C$. Changing the smooth metric on the divisor bundle changes the residual only by $\mathrm{dd}^{\mathrm{c}}$ of a smooth function, so this proves the obstruction. Moreover, $\varphi $ is finite on most of each fiber over $E$ and equals $-\infty $ at the point $o$ of that fiber, so it does not descend, even up to a bounded function. $\square$

Proof. As in example 18.39.20, the norm identity writes $R^ p+T^ b$ over the common denominator for $\mathcal O(p,b)$ on $\mathbb P^ q\times \mathbb P^1$, pulled back to $X$, and its numerator is a sum of squared moduli of holomorphic coefficients. The lower bound for the complex Hessian of the logarithm of such a sum, together with the extension theorem across the pole set, gives

Thus $\omega _ c+\mathrm{dd}^{\mathrm{c}}\varphi _ c\geq c\pi ^*\gamma +\omega _{\mathbb P^1}\geq 0$.

Hartogs' theorem, as in that example, shows that every global section of $A_ e^ m$ can be written uniquely as $s_ E^{em}\rho ^*s$ with $s\in H^0(\mathbb P^ q,\mathcal O(km))$. If $s$ vanishes to order $j$ at $x$, this section vanishes to order $em+j$ generically along $E$. In a chart of the pole set, $\varphi _ c=(c/p)\log (|z|^{2p}+|w|^{2b})+O(1)$. The argument with a unit coefficient in the proof of theorem 18.39.8 and lemma 18.39.7 require

Conversely, this divisorial order gives integrability in every chart of the pole set, and $\varphi _ c$ is locally bounded elsewhere. The vanishing order of $s$ at $x$ equals the generic vanishing order of $\rho ^*s$ along $E$, as the blowup coordinates of example 18.39.17 show. This proves the formula for $V_ m(c)$.

For $0\leq \delta \leq k$, the Poincaré–Lelong formula gives

The non-pluripolar measure does not charge $E$, and it equals the top power of the smooth residual $(k-\delta )\alpha +\delta \gamma $. The class of this residual is $k\rho ^*c_1(\mathcal O(1))-\delta [E]$. The mixed intersection numbers vanish and $\int _ B[E]^ q=(-1)^{q-1}$, as computed in example 18.39.17, so its mass is $k^ q-\delta ^ q$. The divisorial ideal is $\mathcal I(m\psi _ d)=\mathcal O_ B(-\lfloor md\rfloor E)$. Since $em$ is an integer, $\lfloor md\rfloor =em+\ell _ m$, and the condition on $s$ is exactly $j\geq \ell _ m$. This gives the formula for $W_ m(c)$, and the inclusion follows from $\ell _ m\geq v_ m$. If $c\leq e$, both point orders are zero.

For an integer $0\leq j\leq km+1$, homogeneous polynomials of degree $km$ realize every jet of total degree less than $j$ at $x$, and there are $\binom {j+q-1}{q}$ such jets. Hence

For $e<c<e+k$ we have $0\leq v_ m\leq \ell _ m\leq km$ and $\ell _ m-v_ m\leq \lceil a\rceil $, and subtraction gives the exact defect. Each additional jet degree contributes $O(m^{q-1})$ monomials, which gives the bound, and $v_ m=m(c-e)+O(1)$ gives the leading term of $\dim V_ m(c)$.

At $c=e+k$ we have $v_ m=\max \{ km-\lceil a\rceil ,0\} $. For large $m$, the allowed monomials have vanishing orders $km-\lceil a\rceil ,\ldots ,km$ at $x$, and each of these $\lceil a\rceil +1$ degrees has $k^{q-1}m^{q-1}/(q-1)!+O(m^{q-2})$ monomials, which proves the asymptotic formula. The residual is then $k\gamma $, whose top power vanishes. For $c>e+k$, a nonzero polynomial of degree $km$ vanishes at $x$ to order at most $km$. Hence $V_ m(c)=0$ exactly when $v_ m>km$, which is equivalent to $m(c-e-k)-a\geq 1$. $\square$

Proof. We may discard the points with $j_ i=0$; if no point remains, the target is zero. Choose an affine chart containing all the remaining points, and use the corresponding trivialization of $\mathcal O(d)$. For each pair $i\ne k$ choose an affine linear function $L_{ik}$ vanishing at $x_ k$ and nonzero at $x_ i$, and set

Given a jet $b_ i$ at $x_ i$, divide it by the invertible jet of $Q_ i$ at $x_ i$, and let $T_ i$ be the Taylor polynomial of degree at most $j_ i-1$ of the quotient. Then $Q_ iT_ i$ has the jet $b_ i$ at $x_ i$ and vanishes to order at least $j_ k$ at every other point $x_ k$, and its degree is at most $\sum _ kj_ k-1\leq d$. Hence $\sum _ iQ_ iT_ i$, homogenized to degree $d$, realizes all the prescribed jets. In $q$ variables there are $\binom {j_ i+q-1}{q}$ monomials of total degree less than $j_ i$, which gives the dimension formula. $\square$

Proof. The map $(\rho ,\tau _1,\tau _2)$ is an embedding: away from the exceptional divisors $\rho $ is already an embedding, and along $E_ i$ the map $\tau _ i$ separates the tangent directions of $E_ i$ while $\rho $ detects the normal direction. Hence $\kappa >0$. The identity for the quotient metric is the one-point identity of example 18.39.20, applied at each center. For each $i$, the calculation over a common denominator writes $R_ i^2+T^3$ as a sum of squared moduli of holomorphic coefficients over the metric denominator for $\mathcal O(2,3)$. Together with the extension theorem across the pole set, this gives

It follows that $\omega +\mathrm{dd}^{\mathrm{c}}\varphi \geq 4\pi ^*\beta +\omega _{\mathbb P^1}\geq 0$.

By the Poincaré–Lelong formula,

The non-pluripolar product does not charge the divisors and equals $\beta ^ q$. The class of $\beta $ is $2H-[E_1]-[E_2]$, where $H=\rho ^*c_1(\mathcal O(1))$. As in example 18.39.17, $H^ q=1$, the mixed intersection numbers involving $H$ and some $[E_ i]$ vanish, and $[E_ i]^ q=(-1)^{q-1}$; moreover, the two exceptional divisors are disjoint. Expanding gives $\int _ B\beta ^ q=2^ q-2$.

Let $C$ be the strict transform of the line through $x_1$ and $x_2$. Both maps $\tau _ i$ are constant on $C$, so $\int _ C\beta =0$. By Stokes' theorem, $\int _ C(\beta +\mathrm{dd}^{\mathrm{c}}u)=0$ for every smooth real function $u$, whereas a strictly positive form has positive integral over $C$. This proves the obstruction, although the top integral is positive.

The selecting potential satisfies $\varphi \geq \pi ^*\psi $. Its pole set is the disjoint union of the sets $E_ i\times \{ o\} $. Near either of them the other summand is smooth, and

Thus $c_ i=p_ i=2$, $b_{i1}=3$ and $a_ i=2/3$. By theorem 18.39.13, corollary 18.39.19, $W_ m=H^0(B,A^ m(-2m(E_1+E_2)))$, and $V_ m$ is contained in $H^0(B,A^ m(-(2m-1)(E_1+E_2)))$; for an integer $\ell $, the strict local integrability condition $\ell +1>2m-2/3$ means $\ell \geq 2m-1$. Conversely, by lemma 18.39.7, divisibility by these orders gives integrability in every chart of the pole set, and off the pole set the selecting potential is locally bounded. This proves equality for $V_ m$.

Dividing a section of $A^ m$ by $s_1^ ms_2^ m$ gives a section of $\mathcal O(2m)$ away from the two centers, which extends across them by Hartogs' theorem since $q\geq 2$. The pullback of the extension, multiplied by $s_1^ ms_2^ m$, agrees with the original section everywhere, by the identity theorem. If the extended section vanishes to order $j_ i$ at $x_ i$, the corresponding section of $A^ m$ vanishes to order $m+j_ i$ generically along $E_ i$. The two divisorial conditions therefore give exactly the vanishing orders at $x_1$ and $x_2$ displayed in the statement. Since the two point ideals have disjoint supports, their product equals their intersection.

Apply lemma 18.39.24 with $d=2m$ and orders $(m,m)$ or $(m-1,m-1)$; in both cases their sum is at most $d+1$. This gives

Pascal's identity gives the exact defect, and the leading terms give both growth laws. For $m=1$ the vanishing orders for $V_1$ are zero, and the defect is already two. For $q=2$ the dimensions are $\dim W_ m=(m+1)^2$ and $\dim V_ m=m^2+4m+1$. $\square$