In example 18.39.20 the fixed divisor $E$ absorbs the local integrability conditions. If the coefficient of the selecting potential increases, the selected sections must eventually also vanish at the point that was blown up. The following family shows when this happens, and when the mass of the reference potential vanishes.
Keep the blowup $\rho :B\to \mathbb P^ q$ with $q\geq 2$, its exceptional divisor $E$, the extended projection $\tau $, the forms $\alpha ,\gamma ,\kappa $, and the quotient metric on $\mathcal O_ B(E)$ with curvature $\eta $ from that example, and write $x=\rho (E)$ for the point blown up. Let $k,e\geq 1$ be integers and equip
\[ A_ e=\rho ^*\mathcal O_{\mathbb P^ q}(k)\otimes \mathcal O_ B(eE), \qquad \theta =k\alpha +e\eta \]
with the product metric. Let $X=B\times \mathbb P^1$ with its projection $\pi $, let $o\in \mathbb P^1$, and let $R=\| \pi ^*s_ E\| ^2$ and $T=\| s_ o\| ^2$ as before. For positive integers $p,b$ and a real number $c>0$, put
\[ \varphi _ c=\frac cp\log (R^ p+T^ b),\quad \omega _ c=c\pi ^*\kappa +(cb/p+1)\omega _{\mathbb P^1},\quad a=\frac pb, \]
and let $V_ m(c)\subseteq H^0(B,A_ e^ m)$ be the space selected by $\mathcal I(m\varphi _ c)$, regarded on $B$ by descent.
Proposition 18.39.22.F3005ECDDA69 For every $c>0$ we have $\varphi _ c\in \mathrm{PSH}(X,\omega _ c)$, and the selected space is
\[ \begin{gathered} V_ m(c)=s_ E^{em}\rho ^*H^0\bigl (\mathbb P^ q,\mathcal O(km)\otimes \mathfrak m_ x^{v_ m}\bigr ),\\ v_ m=\max \{ \lfloor m(c-e)-a\rfloor ,0\} ,\qquad m\geq 1, \end{gathered} \]
where $\mathfrak m_ x$ is the ideal sheaf of $x$. For $0<c\leq e+k$ put
\[ d=\max \{ c,e\} ,\quad \delta =d-e,\quad \psi _ d=d\log \| s_ E\| ^2,\quad \ell _ m=\lfloor m\delta \rfloor . \]
Then $\psi _ d\in \mathrm{PSH}(B,\theta )$ and
\[ \begin{gathered} \mathrm{MA}_\theta (\psi _ d)=\bigl ((k-\delta )\alpha +\delta \gamma \bigr )^ q, \qquad \int _ B\mathrm{MA}_\theta (\psi _ d)=k^ q-\delta ^ q. \end{gathered} \]
The reference spaces satisfy
\[ \begin{split} W_ m(c)& :=H^0\bigl (B,A_ e^ m\otimes \mathcal I(m\psi _ d)\bigr )\\ & =s_ E^{em}\rho ^*H^0\bigl (\mathbb P^ q,\mathcal O(km)\otimes \mathfrak m_ x^{\ell _ m}\bigr ) \subseteq V_ m(c). \end{split} \]
If $0<c\leq e$, both spaces equal $H^0(B,A_ e^ m)$ for every $m$. If $e<c<e+k$, the defect and the leading term of the dimension are
\[ \begin{gathered} \dim V_ m(c)-\dim W_ m(c) =\binom {\ell _ m+q-1}{q}-\binom {v_ m+q-1}{q}=O(m^{q-1}),\\ \dim V_ m(c)=\frac{k^ q-(c-e)^ q}{q!}m^ q+O(m^{q-1}), \end{gathered} \]
where $\binom {q-1}{q}=0$. At $c=e+k$ the reference $\psi _ d$ has zero mass and
\[ \dim V_ m(e+k) =\frac{(\lceil a\rceil +1)k^{q-1}}{(q-1)!}m^{q-1}+O(m^{q-2}). \]
If $c>e+k$, the selected space is zero exactly when
\[ m(c-e-k)\geq a+1; \]
in particular, it is zero for every $m\geq \lceil (a+1)/(c-e-k)\rceil $.
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
\[ \mathrm{dd}^{\mathrm{c}}\varphi _ c\geq -c\pi ^*\alpha -(cb/p)\omega _{\mathbb P^1}. \]
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
\[ em+j+1>mc-a, \quad \hbox{equivalently}\quad j\geq \max \{ \lfloor m(c-e)-a\rfloor ,0\} . \]
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
\[ \theta +\mathrm{dd}^{\mathrm{c}}\psi _ d=(k-\delta )\alpha +\delta \gamma +d[E]\geq 0. \]
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
\[ h^0\bigl (\mathbb P^ q,\mathcal O(km)\otimes \mathfrak m_ x^ j\bigr ) =\binom {km+q}{q}-\binom {j+q-1}{q}. \]
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 $m(c-e-k)-a\geq 1$.
$\square$
For $0<c<e+k$ the reference $\psi _ d$ has positive mass and the defect is $O(m^{q-1})$, so this family admits the approximation asked for in 24.10.2. At $c=e+k$ the reference mass vanishes and the growth degree of the selected spaces drops to $q-1$; beyond this threshold the selected spaces eventually vanish.
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