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I'd like to believe that this problem has a positive answer, but I don't know a nice reference. Actually I've never worked with embedded curves, so I apologize in advance if the question is too silly.

All schemes are considered to be of finite type over a perfect field $k$. Let $X$ be a connected regular $k$-scheme (not necessarily proper over $k$). For $n>1$ consider the projective space $\mathbb{P}^n_X = \mathbb{P}^n\times_k X$ and let $C$ be a curve in $\mathbb{P}^n_X$, i.e. an integral $1$-dimensional scheme with a closed embedding $\iota\colon C\to \mathbb{P}^n_X$. Let $C^N$ be the normalization of $C$ and let $\phi\colon C^N\to C$ be the normalization morphism.

QUESTION: is there a positive integer $N$ such that

1) $C^N$ embeds in $\mathbb{P}^n_X\times_k \mathbb{P}^N_k = X\times_k \mathbb{P}^n_k\times_k \mathbb{P}^N_k$.

2) the following diagram commutes: \begin{array}{ccc} C^N & \to^{} & X\times_k \mathbb{P}^n_k\times_k \mathbb{P}^N_k \\\ \downarrow^{\phi} & & \downarrow^{\pi}\\\ C & \to^{\iota} & X\times_k \mathbb{P}^n_k \end{array} where the right vertical morphism is (up to reordering the variables) the projection.

Heuristic understanding of the situation: I'm thinking the normalization as the result of successive blow-ups of the original curve, and each blow-up is obtained by attaching a $\mathbb{P}^n$ at the singular point $P$ (the $\mathbb{P}^n$ of the directions of lines passing through $P$) that one uses to separate the branches of $C$ (at least in the case of simple node this description should be accurate). The blow-down morphism is then obtained by contracting all the "extra" $\mathbb{P}^n$.

So, if you like, the problem can be rephrased in the following way: is there a nice way for writing a desingularization for embedded curves by "adding enough variables" to the ambient space?

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    $\begingroup$ Is your $X$ finite type? If $X$ is a finite type $k$-scheme, there should be such a morphism. If $X$ is not finite type (i.e., not quasi-compact), I am pretty sure there are counterexamples, roughly the "strongly projective" versus "weakly projective" problem. $\endgroup$
    – Jason Starr
    Commented Feb 26, 2014 at 10:53
  • $\begingroup$ @JasonStarr Yes, everything is of finite type over the base field $k$. $\endgroup$
    – FedeB
    Commented Feb 26, 2014 at 10:55
  • $\begingroup$ I recommend you look up "finiteness of integral closure" in a commutative algebra textbook -- Eisenbud deduces this as a corollary of the Noether normalization theorem. Then you might look at the exercises in Hartshorne, Chapter III around the Chevalley theorems regarding affiness and ampleness being stable for finite surjective morphisms (such as $\phi$). $\endgroup$
    – Jason Starr
    Commented Feb 26, 2014 at 10:59
  • $\begingroup$ Of course $\phi$ is finite and surjective (all schemes are excellent here) and one can embed $C^N$ in some projective space over $X$, in general. My question is about the possibility of realizing this embedding so that the resulting morphism $\mathbb{P}^M_X\to\mathbb{P}^n_X$ is given by the actual projection on the first factors. $\endgroup$
    – FedeB
    Commented Feb 26, 2014 at 11:03
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    $\begingroup$ "My question is about ..." Yes, of course. Denote by $\kappa:C^N\to \mathbb{P}^N_X$ any closed immersion of $X$-schemes. Then the morphism $(\iota\circ \phi,\kappa):C^N\to \mathbb{P}^n_X\times_X \mathbb{P}^N_X$ is also a closed immersion. $\endgroup$
    – Jason Starr
    Commented Feb 26, 2014 at 13:16

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