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After reading Fulton's book "Algebraic Curves", I know how to do resolution of singular points on curves. Given an affine equation, I can get it's non-singular affine model, i.e the normalization of its affine coordinate ring. The problem is that in Fulton's book, he worked with algebraically closed field and he worked with the origin (0,0) also. If one do blowing-up (p.165 in Fulton's book) of a curve defined a field $k$ which is not necessary algebraically colsed and with a singular point which is not necessary $k$-rational, one get an eqaution which is defined over a finite extension of the field $k$. But for any affine curve (i.e integral scheme of dimension 1), one can form the normalization of its coordinate ring and hence get a non-singular affine curve which is birational to the original curve. My question is that how one can get the equation for this normalized affine curve? I know there are algorithms to compute the integral closure of function fields, but the result is an integral basis rather that an equation. Moreover, I would like to know if there is a systematic method to get this normalized affine equation with the blowing-up method.

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It won't be possible in general to get a single equation, because the curve does not necessarily admit a nonsingular plane model (globally). It is possible to get equations using Computer Algebra Systems, for instance Singular does it as explained here.

Concerning blowups: if the curve has a singular point which is not rational, then all of its conjugate points will be singular too, and they'll form a scheme that is defined over $k$. I never had to deal with such examples, but if $k$ is perfect I suppose that the blowup centered at that scheme will simplify the singularity and, iterating, eventually resolve it.

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Could you say more about "blowup centered at that scheme" or give a reference for this? Because I learned blow up by Fulton's book, which is blowing-up at (closed and rational) points only. Thanks. –  user565739 May 31 '11 at 15:12
    
Hartshorne explains the general construction via the (sheaf of) graded algebras associated to a (sheaf of) ideals in II.7. There must be a simpler exposition for the affine case written somewhere (to avoid the formalism of sheaves), but unfortunately I don't know a reference. :( Sorry! –  quim May 31 '11 at 16:08
    
The rough idea is as follows (though I recommend to get a real explanation in full, if someone can give a better reference). Let ${\mathfrak p}\subset R$ be the ideal of the scheme at which you want to blow up (I assume R=k[x,y]). Let A be the graded algebra $\bigoplus {\mathfrak p}^n$. Then the blowup is a variety, projective over R, associated to the graded algebra A. To obtain its equations in $\mathbb{P}^n_R$, you need to know a minimal set of generators (n+1 of them) and their relations. –  quim May 31 '11 at 16:15
    
Just a quick comment, the algebra quim describes is called the Rees algebra. It might help you find references where some examples are worked out. –  Karl Schwede Jun 2 '11 at 0:59
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