Let $X = Z(xy - zw)$ in $\mathbb{A}^4$ and $Y = Z(x, z)$. If $\pi: B \rightarrow \mathbb{A}^4$ is the blow up of $Y$, then how can I find the strict transform of $X$ and the exceptional divisor?
I honestly have no idea how to find the strict transform in this case. I'm only fairly comfortable with blowing up single points. This is what I have so far, gathered from random lecture notes. It only deals with the exceptional divisor and is rather messy:
Let $E = \pi^{-1}(Y)$ and $J = I(Y) = (x, z)$. By Theorem 14.7, $B(J) = U_1 \cup U_2$ where $U_1, U_2$ are affine and $\mathcal{O}_{B(J)}(U_1) = k[\mathbb{A}^{4}][z/x] = k[x, y, z/x]$, $\mathcal{O}_{B(J)}(U_2) = k[\mathbb{A}^{4}][x/z] = k[y, z, x/z].$ We have $x = 0$ is a local equation of $E$ in $U_1$, $z = 0$ is a local equation of $E$ in $U_2$, and $k[E \cap U_1] = k[x, y, z/x]/(x)k[x, y, z/x] = k[y, z/x]$, $k[E \cap U_2] = k[y, z, x/z]/(x)k[y, z, x/z] = k[y, x/z].$ (I do not recognize what this means about $E$, but $E \not = \mathbb{A}^{2}$. Random guess: $E$ is the subset of $\mathbb{P}^{2}$ with coordinates $x, y, z$ described by $(x : xy : z)$ for some $(x, y, z) \in U_1$ and $(x : yz : z)$ for some $(x, y, z) \in U_2$. Using $U_1 \cap E = Z(J\mathcal{O}_{B(J)}(U_1)) = Z(x)$ and $U_2 \cap E = Z(J\mathcal{O}_{B(J)}(U_2)) = Z(z)$, we get $E = \{(x : 0 : z) \mid x \in \pi_1(U_1), z \in \pi_3(U_2)\}$ . . . )