Suppose $X$ is a smooth, projective curve, $Y$ is an arbitrary curve(may be singular), and both curves are over an algebraically closed field $k$ with character 0. Let $f: X \to Y$ be a morphism between curves. Is $f$ a projective morphism? Here, projective morphism is in the sense of Hartshorne, i.e. $X \to Y$ factors through $X \to \mathbb{P}^{n}_{Y} \to Y$, with $X \to \mathbb{P}^{n}_{Y}$ a closed embedding, $\mathbb{P}^{n}_{Y} \to Y$ the the natural projection to $Y$ factor.
I guess it is projective by the following general heuristic argument:
Statement:Suppose $X \subset \mathbb{P}^{n}$ is a closed subvariety, $Y$ is another variety, then any morphism $f: X \to Y$ is projective.
One can define $f' :X \to \mathbb{P}^{n}_{Y}$ by $x \mapsto (x,f(x))$, and this is an injective map. Moreover, because $X$ is proper, its image must be closed. I guess these guarantee $f'$ is a closed embedding, and the projection $\mathbb{P}^{n}_{Y} \to Y$ is easy to define.
I am not quite sure about the above argument, especially $f'$ being a closed embedding. Any suggestions?