Question

Suppose $X$ is a projective variety over a field $k$ with function field $FF(X)$. Let $\widetilde{X}$ be the normalization of $X$ in a finite extension of $FF(X)$ (for the definition of normalization in finite field extension, please see class41 of ravi vakil's notes here).

I know $\widetilde{X}$ is also projective, because normalization is finite hence projective and composition of projective morphism is projective. But I want to know if there is a simple procedure to embed $\widetilde{X}$ in a projective space explicitly (by simple I mean the dimension of projective space is as small as possible, by explicit I mean to write down the equations explicitly). I am particularly interested in the case when $X$ is a curve.

Thanks in advance.

Answer 1

If X is a curve, a good way is certainly to construct the normalization iteratively by blowing-up the singular points : you will blow-up the ambient projective space as well and it is very easy to embed the blow-up of a projective space in a projective space. Stop when you get a smooth curve. You obtain a finite morphism $Y\rightarrow X$, with $Y$ smooth, and thus normal. Since the morphism is finite, $Y$ is the normalization. A good reference is the book of Kollár, Lectures on Resolution of Singularities.