Lemma 18.39.24.822490631827 Let $x_1,\ldots ,x_ N$ be distinct points of $\mathbb P^ q$, $q\geq 1$, and let $d,j_1,\ldots ,j_ N$ be nonnegative integers with $\sum _ i j_ i\leq d+1$. Then the map from $H^0(\mathbb P^ q,\mathcal O(d))$ to the jets of order less than $j_ i$ at the points $x_ i$ is surjective. In particular,
\[ h^0\left(\mathbb P^ q,\mathcal O(d)\otimes \bigcap _ i\mathfrak m_{x_ i}^{j_ i}\right) =\binom {d+q}{q}-\sum _ i\binom {j_ i+q-1}{q}, \]
where $\binom {q-1}{q}=0$.
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
\[ Q_ i=\prod _{k\ne i}L_{ik}^{j_ k}. \]
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.
$\square$
Pome
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