Let $X \to Y$ be a projective morphism. So, this map factors through a closed immersion $i$ of $X$ into $\mathbb{P}^n \times Y$ for some $n$ followed by the projection map to $Y$. When is it possible to define a map $\phi$ from $Y$ to $\mathbb{P}^n \times Y$ such that the image of $\phi$ is contained in the image of the closed immersion $i$ such that $\phi$ composed with the projection map is identity? When can we say that $\phi$ is itself a closed immersion.
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