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.