A question on very ample line bundle and closed immersion

Question

Let $X$ be a projective scheme, $i:X \hookrightarrow \mathbb{P}^n$ a closed immersion, $\mathcal{L}:= i^*\mathcal{O}_{\mathbb{P}^n}(1)$ a very ample line bundle. Let $j:{\mathbb{P}^n} \hookrightarrow \mathbb{P}^N$ be a closed immersion and $\mathcal{L}':= (j \circ i)^* \mathcal{O}_{\mathbb{P}^N}(1)$. Is it true that $\mathcal{L}\cong \mathcal{L}'$?

Answer 1

It need not be true in general. For example, take $X = \mathbb{P}^1$, $n=1$, $i$ the identity map, $N = 2$ and $j$ the degree 2 Veronese map into $\mathbb{P}^2$. In this situation $\mathcal{L} \cong \mathcal{O}_{\mathbb{P}^1}(1)$ but $\mathcal{L}' \cong \mathcal{O}_{\mathbb{P}^1}(2)$.