This is exercise 1.10 from Reid's Young person's guide to canonical singularites.

Let $X=\mathbb{C}^3/ \mu_3$ where $\epsilon \in \mu_3$ acts by $$ (x,y,z) \to (\epsilon x, \epsilon y, \epsilon^2 z).$$ Then blowing up the origin gives us $E_1 \cup E_2$ where $E_1$ is a plane and $E_2$ is a quartic scroll.

We know $X=\text{Spec}(A)$ where $$A = \mathbb{C}[x^3, x^2 y, x y^2, y^3, xz , yz, z^3]= \mathbb{C}[u_0, u_1, u_2, u_3, v_0, v_1, w]/I$$ $$I=<u_0u_3-u_1 u_2, u_0u_2-u_1^2, u_1u_3-u_2^2, u_2 v_1- u_3 v_0, u_1 v_1 - u_2 v_0, u_0 v_1- u_1 v_0, \\ v_0^3 - u_0 w, v_0^2 v_1 - u_1 w, v_0 v_1^2 - u_2 w, v_1^3 - u_3 w>$$

If I consider the map $$(s, t, p) \to (s, st , st^2, st^3, sp, stp, s^2p^3),$$ it seems that $s=0$ corresponds to $E_1$ and the pullback of $$s = \frac{1}{u_0^5}(du_0 \wedge du_1 \wedge du_2)^{\otimes 3 }$$ has zeros of order 1 along $E_1$.

I'm not sure how to get $E_2$. If I blow up the origin of $\mathbb{C}[u_0, u_1, u_2, u_3, v_0, v_1, w],$ how can I identify $\mathbb{P}^6$ with $E_1 \cup E_2$? Why the pullback of $s$ has has zeros of order 2 along $E_2$.