I am trying to desingularize the curve $V=V(y^n-x^d)$, where $n>d$ and $gcd(n,d)=g$. It is singular at $P(0,0)$ so I am trying a blow-up.
My intuition is that a blow-up is locally like $(x,y)\mapsto (xy, y)$ or $(x,y)\mapsto (x, xy)$. So I will obtain still singular curves: $y^n-(xy)^d$ that is $y^{n-d}-x^d$, and in the same way I will arrive to $y^{n-d_1d}-x^d$ with $d_1$ the entire divisor of $d$ and $n$, so $n-d_1 d=r_1$ is the rest. Like in the Euclid's algorithm, I will come to something like $y^g-1$.
I am stuck and I need some help as a hint or a reference
$g=\gcd(n,d)$ and $n>d>0$.
The blow up of $V(y^n-x^d)$ at $(0,0)$
(the Zariski closure of $\{((x,y), [u:v]), xu+yv=0,y^n-x^d=0,(x,y)\ne (0,0)\}$)
is isomorphic to $V(Y^{n-d}-X^d)$ (where $(x,y)=(XY,X)$)
repeating several times we'll arrive at $V(W^g-Z^g)$.
(in characteristic $0$) $V(W^g-Z^g)$ is smooth if $g=1$, otherwise it is an union of several smooth curves intersecting at $(0,0)$.