I met a problem when reading Positivity In Algebraic Geometry I. It is Example $1.8.15$ in Chpater 1 Section 8, which is called Green's Theorem.
Let $W$ be a subspace of $H^{0}(P, O_{P}(d))$, and $W$ is a codimension-c vector space, which can be treated as a linear series.
Now assume that $P:= P(V)$, $V$ is a vector space. Then the example shows 2 exact short sequences of locally free sheaves.
The first one is the evaluation map from $S^{d}\otimes O_{P}$ to $O_{P}(d)$. It is clear that this map is surjective by checking locally.
The Second is confusing to me. It states that similar to the first one, we also have a surjective morphisim :
$W\otimes O_{P} \to O_{P}(d)$, since $W$ is just a proper subspace of $H^{0}(P, O_{P}(d))$, I do not think this surjectivity is as obvious as the former one. But I do not know how to check it.
I wish someone could help to explain it to me, thank you very much!
Since $O_P(d)$ is a line bundle, this map is surjective as long as, for each point $p \in \mathbf P(V)$, there is an element of $W$ which is nonzero at $p$. But that is precisely what the word "free" in the hypothesis "free linear series" means.