Sorry for the easy question but this is driving me crazy. Consider the blowing up of the curve $(y^2-x^3)^2+y^5$ at the origin.
On the first blowing up, on the chart that intersects the exceptional divisor I have: $x^4(y^4-2xy^2+x^2+xy^5)$.
The second blowing up, on the chart that intersects the exceptional divisor: $y^2(y^4x+y^2-2xy+x^2)$
On the third blowing up, on the chart that intersects the exceptional divisor: $x^2(x^3y^4+y^2-2y+1)$.
So the last strict transform is smooth at $(0,0)$ (the same for the other chart). I naively thought that this is the end, and I solved the singularity.
However, the multiplicity sequence for the resolution is (4,2,2,2,1,1) , and the dual graph of the resolution is like a T, and it has six exceptional divisors. (the dual graph is here link text) So what am I missing? I tried several examples, and I am running into trouble specially when there is a node on my way. How can I keep going with the resolution all the way until the end?
Ps: Calculations like the multiplicity sequence were done in Singular for avoiding trivial mistakes.