Quartic in the projective space and exact sequence of sheaves

Question

I am trying to understand a few basics about the twisting sheaves. I read that, given a smooth quartic $S$ in $\mathbb P^3$, we have an exact sequence $0\rightarrow\mathcal O_{\mathbb P^3}(-4)\rightarrow\mathcal O_{\mathbb P^3}\rightarrow\mathcal O_S\rightarrow 0$. What is the map $\mathcal O_{\mathbb P^3}(-4)\rightarrow\mathcal O_{\mathbb P^3}$? I suppose the map $\mathcal O_{\mathbb P^3}\rightarrow\mathcal O_S$ is just the restriction of the sheaf?

Thank you

Answer 1

Question: "I suppose the map $O_{P^3}→O_S$ is just the restriction of the sheaf? Thank you"

Answer: If $F\in T:=k[x_0,..,x_3]$ is the degree $4$ homogeneous polynomial defining $S:=Z(F)$ you get an exact sequence

$$0 \rightarrow T(-4) \xrightarrow{\phi} T \rightarrow T/(F) \rightarrow 0$$

of graded $T$-modules where $\phi(a):=af$, inducing the exact sequence

$$0 \rightarrow \mathcal{O}_{\mathbb{P}^3}(-4) \rightarrow \mathcal{O}_{\mathbb{P}^3} \rightarrow \mathcal{O}_S \rightarrow 0$$