I'm reading from the book "Geometry of algebraic curves", by Griffiths, Harris, Arbarello and Cornalba.
In the middle of page 5 they define the map $\phi_{\mathscr{D}}:C\to \mathbb{P}V^*$, from a curve $C$ to the projectified linear subspace $\mathbb{P}V$ of $H^0(C,L)$, by the prescription $\phi(p)=$"sections $s\in V$ which vanish at $p$".
It doesn't make sense to me! It should be defined, instead, as
$\phi(p)=$"sections $s\in V$ which don't vanish at $p$",
so that the target is really $\mathbb{P}V^*$, since the zero section doesn't belong to the image of any point.
Do you agree with me and this is a typo or am I losing something?
Here's part of the page:
My guess would be that $p\in C$ defines a linear functional on $V$ by the rule $p(s) = s(p).$ We see that it is linear since we of course have $p(s_1+s_2)=(s_1+s_2)(p)=s_1(p)+s_2(p)$ and similarly for scalar multiplication. Thus, the set $\{s\in V:s(p)=0\}$ is a hyperplane in $V$ cut out by the linear functional $p.$ That is, we map $p$ to the point in $\mathbb P(V^*)$ corresponding to the hyperplane.