Proposition 18.37.3.BEF6F7AD320C Pullback identifies the selected space $H^0(X,L^ m\otimes \mathcal I(m\varphi ))$ with
\[ V_ m=H^0\left(B,A^ m\otimes \bigcap _{b\in Z}\mathfrak m_ b^{r_{m,b}}\right). \]
The space
\[ W_ m=H^0\left(B,A^ m\otimes \bigcap _{b\in Z}\mathfrak m_ b^{s_{m,b}}\right) \]
satisfies $W_ m\subseteq V_ m$ and
\[ 0\leq \dim V_ m-\dim W_ m \leq \sum _{b\in Z}\sum _{j=r_{m,b}}^{s_{m,b}-1} \binom {j+q-1}{q-1} =O(m^{q-1}), \]
where an empty inner sum is zero. The implied constant depends only on the points and their coefficients, not on $m$.
Proof.
Lemma 5.37.1, applied with a point as center, gives
\[ \mathcal I(m\varphi )_{p_ j}=\mathfrak m_{p_ j}^{\max \{ \lfloor ma_ j\rfloor -n+1,0\} }; \]
away from the points $p_ j$ the ideal is the full structure sheaf.
Every section of $L^ m$ descends to $B$, by the argument in proposition 18.36.1. Near $p_ j$ choose submersion coordinates $(z,\tau )$ with $\pi (z,\tau )=z$. A pulled-back section has a local coefficient $g(z)$ that does not depend on $\tau $, and its vanishing order at $(0,0)$ equals the vanishing order of $g$ at $z=0$. The conditions imposed by the points of one fiber are therefore vanishing conditions at the same base point, and the strongest of them is vanishing to order $r_{m,b}$. This proves the identification.
For the dimension bound, map each section in $V_ m$ to its jets modulo $\mathfrak m_ b^{s_{m,b}}$ at the finitely many points $b\in Z$. The kernel of this map is $W_ m$, so local trivializations of $A^ m$ give an injection
\[ V_ m/W_ m\hookrightarrow \bigoplus _{b\in Z}\mathfrak m_ b^{r_{m,b}}/ \mathfrak m_ b^{s_{m,b}}. \]
Surjectivity of the jet map is not needed. In $q$ variables there are $\binom {j+q-1}{q-1}$ monomials of degree $j$, which gives the dimensions of these quotients. Finally $0\leq s_{m,b}-r_{m,b}\leq f$ and $s_{m,b}=O(m)$, which proves the bound, also when $q=1$ or $f=0$.
$\square$
Pome
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