My question refers to a former theread of mine: Global Sections of Pullback of Tautological Line Bundle
Here the setting is the same, so $C$ is a Gorenstein, integral curve (therefore $1$-dimensional, proper $k$-scheme over a fixed field), $\mathcal{L} \in Pic(C)$ ample, $f: C \to \mathbb{P}^n _k$ a proper morphism induced via $\mathcal{L}:=f^*(\mathcal{O}_{\mathbb{P}^n}(1))$ and the image $C' := f(C) = im(f) \subset \mathbb{P}^n$ an integral curve of $\mathbb{P}^n$.
$f$ factorises as $C \xrightarrow{f'} C' \xrightarrow{\iota} \mathbb{P}^n$. By definition we have $\mathcal{O}_{C'}(1) := \iota^* \mathcal{O}_{\mathbb{P}^n}(1)$.
I'm trying to get the map
$H^0(C',i^*(\mathbb{P}^n(1)) \to H^0(C,\mathcal{L})$
I have done following attempts: To start with following exact(?) sequence:
$$0 \to \mathcal{O}_{C'} \to f'_*(\mathcal{O}_{C'}) \to \mathcal{F} \to 0$$
Here I just suppose that $\mathcal{O}_{C'} \to f'_*(\mathcal{O}_{C'})$ is injective. Can anybody explain me a concrete argument?
Then it seems seductive to tensor it with $\mathcal{O}_{C'}(1)$, so one gets:
$$0 \to \mathcal{O}_{C'}(1) \to f'_*(\mathcal{O}_{C'}) \otimes \mathcal{O}_{C'}(1) \to \mathcal{F} \to 0$$
The right term stays $\mathcal{F}$ since it's skyscraper, since $f'$ dense.
Here my problem is how to see that $f'_*(\mathcal{O}_{C'}) \otimes \mathcal{O}_{C'}(1) \cong f'_*(\mathcal{L})$ holds?
You have some $C$ vs. $C'$ typos currently, but it looks like the thing you're asking about is just the Projection Formula. I'm going to write $g$ instead of $f'.$ We have $$g_*(\mathcal{L}) = g_*(g^*\mathcal{O}_{C'}(1)) \cong g_*(g^*\mathcal{O}_{C'}(1) \otimes \mathcal{O}_C) \cong \mathcal{O}_{C'}(1) \otimes g_*(\mathcal{O}_C).$$