I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities. So the Newton's method gives us a Puiseux series parametrizing the branches of a curve at a point. What I do not understand is how this gives a resolution of singularities. Since the Puiseux series is a power series, where is the variety here? I know this is a basic question, but most of the references skim through this aspect and make it sound self-evident. I am hoping for some simple explanation of the connection. Thanks!
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$\begingroup$ what are these "most of the references" you do not mention? If you tell us where you did look it is easier to give you useful references! $\endgroup$– Mariano Suárez-ÁlvarezMay 18, 2013 at 7:04
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1$\begingroup$ "Lectures on Resolution of Singularities" by Kollar describes the Newton's method, but does not explicitly describe how the Puiseux series provides a resolution of singularities. Another reference by Cutkosky on the same topic mentions it in passing as a corollory of a more complicated result. I was hoping there is a direct explanation in the special case of curve singularities. $\endgroup$– BharatRamMay 18, 2013 at 7:10
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$\begingroup$ The Puiseux series give directly a smooth parametrization (i.e. a Riemann surface) of a singular curve, you can see Theorem 1 in Section 8.3 and the following remark on page 389-390 of Brieskorn's "Plane Algebraic Curves". If you want to realize this smooth model as a variety, Theorem 9 in Section 8.4 of Brieskorn's book which I mentioned before gives you a method by blow-up processes. $\endgroup$– Yuchen LiuMay 18, 2013 at 12:08
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