I'm familiar with blowups in classical algebraic geometry but I'm still learning about blowup of schemes.
For now I'm trying to be as concrete as possible, because the formal definition is incomprehensible to me.
Let's take the example of the cubic $yz^2-x^3=0$ and the triple line $y^3=0$ both in $\Bbb{P}^2=\text{Proj}(\Bbb{C}[x,y,z])$. From this article section, the blowup of $\Bbb{P}^2$ at the $9$ intersection points can be described by: $$\text{Proj}\left(\frac{\Bbb{C}[x,y,z,s,t]}{s(yz-x^3)+ty^3}\right)\to \Bbb{P}^2$$
To be more concrete, I'm trying to describe each of those $9$ blowups in a way analogous to the classical blowup.
Here's my attempy for the first blowup:
In ${z\neq 0}$, we have a cubic $C_1=V(y-x^3)$ and the triple line $C_2=V(y^3)$ in $\Bbb{A}^2$. The blowup surface $X$ would be a subscheme of $\Bbb{A}^2\times \Bbb{P}^1$ defined by $sy=tx$. Let $U:=\{s\neq 0\}$ and let $\widetilde{C_1}, \widetilde{C_2}$ be the strict transforms. Then:
\begin{align*} \widetilde{C_1}\cap U &=V(y-x^3,y-tx)\\ &=V(tx-x^3,y-tx)\\ &= V(x,y)\cup V(t-x^2,y-tx)\\ \\ \widetilde{C_2}\cap U &=V(y^3,y-tx)\\ &=V((tx)^3,y-tx)\\ &=V(x^3,y) \cup V(t^3,y-tx) \end{align*}
Taking $x,t$ as local parameters in $X$, then $\widetilde{C_1},\widetilde{C_2}$ and the exceptional divisor are described by $t-x^2=0$, $t^3=0$ and $x=0$ respectively.
This way, $\widetilde{C_2}$ has multiplicity $3$ (a triple curve), but I don't what to make of the $x^3$ that appeared in the last equality above. Does this mean the exceptional divisor has a multiplicity?
Does this make sense at all? Any help will be appreciated, thank you!