Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

1 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.

share|improve this answer
    
How can I know when to stop? Can you point out some references for the algorithm? Thanks! –  Liu Hang Jan 9 '12 at 1:36
    
I completed my answer with your questions. Please ask if there is still something unclear. –  Lierre Jan 9 '12 at 9:56

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.