**Proof.**
It is clear that (1) implies (2). It is also clear that (4) implies (1); the hypothesis of quasi-separation in (4) is used to guarantee that $f_*\mathcal{L}$ is quasi-coherent via Schemes, Lemma 26.24.1.

Assume (2). We will prove (4). Let $S = \bigcup V_ j$ be an open covering as in (2). Set $X_ j = f^{-1}(V_ j)$ and $f_ j : X_ j \to V_ j$ the restriction of $f$. We see that $f$ is separated by Lemma 29.38.6 (as being separated is local on the base). By assumption there exists a quasi-coherent $\mathcal{O}_{V_ j}$-module $\mathcal{E}_ j$ and an immersion $i_ j : X_ j \to \mathbf{P}(\mathcal{E}_ j)$ with $\mathcal{L}|_{X_ j} \cong i_ j^*\mathcal{O}_{\mathbf{P}(\mathcal{E}_ j)}(1)$. The morphism $i_ j$ corresponds to a surjection $f_ j^*\mathcal{E}_ j \to \mathcal{L}|_{X_ j}$, see Constructions, Section 27.21. This map is adjoint to a map $\mathcal{E}_ j \to f_*\mathcal{L}|_{V_ j}$ such that the composition

\[ f_ j^*\mathcal{E}_ j \to (f^*f_*\mathcal{L})|_{X_ j} \to \mathcal{L}|_{X_ j} \]

is surjective. We conclude that $\psi : f^*f_*\mathcal{L} \to \mathcal{L}$ is surjective. Let $r_{\mathcal{L}, \psi } : X \to \mathbf{P}(f_*\mathcal{L})$ be the associated morphism. We still have to show that $r_{\mathcal{L}, \psi }$ is an immersion; we urge the reader to prove this for themselves. The $\mathcal{O}_{V_ j}$-module map $\mathcal{E}_ j \to f_*\mathcal{L}|_{V_ j}$ determines a homomorphism on symmetric algebras, which in turn defines a morphism

\[ \mathbf{P}(f_*\mathcal{L}|_{V_ j}) \supset U_ j \longrightarrow \mathbf{P}(\mathcal{E}_ j) \]

where $U_ j$ is the open subscheme of Constructions, Lemma 27.18.1. The compatibility of $\psi $ with $\mathcal{E}_ j \to f_*\mathcal{L}|_{V_ j}$ shows that $r_{\mathcal{L}, \psi }(X_ j) \subset U_ j$ and that there is a factorization

\[ \xymatrix{ X_ j \ar[r]^-{r_{\mathcal{L}, \psi }} & U_ j \ar[r] & \mathbf{P}(\mathcal{E}_ j) } \]

We omit the verification. This shows that $r_{\mathcal{L}, \psi }$ is an immersion.

At this point we see that (1), (2) and (4) are equivalent. Clearly (4) implies (3). Assume (3). We will prove (1). Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ S$-algebras generated in degree $1$ over $\mathcal{O}_ S$. Consider the map of graded $\mathcal{O}_ S$-algebras $\text{Sym}(\mathcal{A}_1) \to \mathcal{A}$. This is surjective by hypothesis and hence induces a closed immersion

\[ \underline{\text{Proj}}_ S(\mathcal{A}) \longrightarrow \mathbf{P}(\mathcal{A}_1) \]

which pulls back $\mathcal{O}(1)$ to $\mathcal{O}(1)$, see Constructions, Lemma 27.18.5. Hence it is clear that (3) implies (1).
$\square$

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