I am studying algebraic geometry from the Gathmann’s notes and I have to resolve exercise (it is not on the notes)
Resolve the singularities of the following curve in $\mathbb{A}^2$ by subsequent blow-ups of the singular points. This means: Compute its singular locus and blow it up. Then replace $\mathcal{C}$ by its strict transform $\tilde{\mathcal{C}}$. Continue this process until the resulting curve is smooth. $$\mathcal{C}=V(x^3-y^5)$$
I am very confused about the computation of the blow up and I would some help for this exercise. It’s not clear to me if to compute strict transform of $\mathcal{C}$ is enough to compute the blow up of $\mathbb{A}^2$ at the singularities and with an affine charts on the blow up, “immerse” $\mathcal{C}$ in $\tilde {\mathcal{C}}$. But in any case, I don’t know how to compute the blow up of $\mathbb{A}^2$ in a finite number of points if the are greater than one.
My idea is to replace the blow up of $\mathbb{A}^2$ in a point $a$ in which we leave all points except $a$ unchanged and we replace $a$ with a projective line. But I have no idea how I can formalise this idea.
Let us first blow the curve up at $(0,0)$. First of all, the blow-up of $\mathbb A^2$ at $(0,0)$ is $$\{(x,y)\in\mathbb A^2,[t_0:t_1]\in\mathbb P^1:xt_1=yt_0\}.$$ It is covered by two open subsets $$U_1:=\{(x,xt_1)\in\mathbb A^2,[1:t_1]\in\mathbb P^1\}\simeq\mathbb A^2$$ and $$U_2:=\{(yt_0,y)\in\mathbb A^2,[t_0:1]\in\mathbb P^1\}\simeq\mathbb A^2.$$ Thus the blow-up $\widetilde C$ of $C$ is such that: $$\widetilde C\cap U_1=\{(x,xt_1)\in\mathbb A^2,[1:t_1]\in\mathbb P^1:1=x^2t_1^5\}\simeq\{(x,t_1)\in\mathbb A^2:1=x^2t_1^5\}$$ and $$\widetilde C\cap U_2=\{(yt_0,y)\in\mathbb A^2,[t_0:1]\in\mathbb P^1:t_0^3=y^2\}\simeq\{(y,t_0)\in\mathbb A^2:t_0^3=y^2\}.$$ Here $\widetilde C\cap U_1$ is non-singular but $\widetilde C\cap U_2$ is not at $(y,t_0)=(0,0)$ must be blown-up again.