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Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process terminates, we need a measurable way of seeing an "improvement" in the singularity.

Define the Milnor number of $f$ as $\mu(f) = dim_{\mathbb{C}} \frac{\mathbb{C}[[x,y]]}{<\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}>}$,

i.e. the dimension of this ring as a complex vector space.

I believe I once heard that we can use the Milnor number of $f$ to resolve singularities, which I suppose should mean that the Milnor number decreases after blowing-up until it's $0$ (where we then have a smooth curve).

Is this true? If so could you provide a reference, and if not a quick counter-example?

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"to make sure this process terminates": do not we know it terminates, by Hironaka's theorem? Also, I do not understand: when you say "the Milnor number decreases", what is the new $\mu(f')$ which you compare to the first $\mu(f)$? Sorry for being pedant but I would like to understand your question better, I find it interesting :) –  Brenin Mar 23 at 22:48
@Brenin, I think Sergio must be wondering how to replace other invariants to prove Hironaka's theorem, i.e., apply induction on the Milnor number. The new Milnor number must be the Milnor number on any singularity of the fibre above the origin. –  Andrew Mar 23 at 23:06
Andrew, this is correct. I already know how to compute Hironaka's invariant (using orders and exceptional divisors), but for curves there are other methods. The new Milnor number is as you say. I basically wanted a simple way of explaining to my student's why blowing-up finitely many times works - the hope being that the Milnor number decreases with each successive blow-up. –  Sergio Da Silva Mar 24 at 1:13

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